NCERT Solutions for Class 11 Science Math Chapter 11 - Conic Sections

Answer:

The equation of a circle with centre (h, k) and radius r is given as

(x­ – h)² + (y ­– k)² = r²

It is given that centre (h, k) = (0, 2) and radius (r) = 2.

Therefore, the equation of the circle is

(x – 0)² + (y – 2)² = 2²

x² + y² + 4 ­– 4 y = 4

x² + y² ­– 4y = 0

Answer:

The equation of a circle with centre (h, k) and radius r is given as

(x­ – h)² + (y ­– k)² = r²

It is given that centre (h, k) = (–2, 3) and radius (r) = 4.

Therefore, the equation of the circle is

(x + 2)² + (y – 3)² = (4)²

x² + 4x + 4 + y² – 6y + 9 = 16

x² + y² + 4x – 6y – 3 = 0

Answer:

The equation of a circle with centre (h, k) and radius r is given as

(x­ – h)² + (y ­– k)² = r²

It is given that centre (h, k) = and radius (r) =.

Therefore, the equation of the circle is

Answer:

The equation of a circle with centre (h, k) and radius r is given as

(x­ – h)² + (y ­– k)² = r²

It is given that centre (h, k) = (1, 1) and radius (r) =.

Therefore, the equation of the circle is

Answer:

The equation of a circle with centre (h, k) and radius r is given as

(x­ – h)² + (y ­– k)² = r²

It is given that centre (h, k) = (–a, –b) and radius (r) =.

Therefore, the equation of the circle is

Answer:

The equation of the given circle is (x + 5)² + (y – 3)² = 36.

(x + 5)² + (y – 3)² = 36

⇒ {x – (–5)}² + (y – 3)² = 6², which is of the form (x – h)² + (y – k)² = r², where h = –5, k = 3, and r = 6.

Thus, the centre of the given circle is (–5, 3), while its radius is 6.

Answer:

The equation of the given circle is x² + y² – 4x – 8y – 45 = 0.

x² + y² – 4x – 8y – 45 = 0

⇒ (x² – 4x) + (y² – 8y) = 45

⇒ {x² – 2(x)(2) + 2²} + {y² – 2(y)(4)+ 4²} – 4 –16 = 45

⇒ (x – 2)² + (y –4)² = 65

⇒ (x – 2)² + (y –4)² = , which is of the form (x – h)² + (y – k)² = r², where h = 2, k = 4, and .

Thus, the centre of the given circle is (2, 4), while its radius is.

Answer:

The equation of the given circle is x² + y² – 8x + 10y – 12 = 0.

x² + y² – 8x + 10y – 12 = 0

⇒ (x² – 8x) + (y² + 10y) = 12

⇒ {x² – 2(x)(4) + 4²} + {y² + 2(y)(5) + 5²}– 16 – 25 = 12

⇒ (x – 4)² + (y + 5)² = 53

, which is of the form (x – h)² + (y – k)² = r², where h = 4, k = –5, and .

Thus, the centre of the given circle is (4, –5), while its radius is.

Answer:

The equation of the given circle is 2x² + 2y² – x = 0.

, which is of the form (x – h)² + (y – k)² = r², where h = , k = 0, and .

Thus, the centre of the given circle is, while its radius is.

Answer:

Let the equation of the required circle be (x – h)² + (y – k)² = r².

Since the circle passes through points (4, 1) and (6, 5),

(4 – h)² + (1 – k)² = r² … (1)

(6 – h)² + (5 – k)² = r² … (2)

Since the centre (h, k) of the circle lies on line 4x + y = 16,

4h + k = 16 … (3)

From equations (1) and (2), we obtain

(4 – h)² + (1 – k)² = (6 – h)² + (5 – k)²

⇒ 16 – 8h + h² + 1 – 2k + k² = 36 – 12h + h² + 25 – 10k + k²

⇒ 16 – 8h + 1 – 2k = 36 – 12h + 25 – 10k

⇒ 4h + 8k = 44

⇒ h + 2k = 11 … (4)

On solving equations (3) and (4), we obtain h = 3 and k = 4.

On substituting the values of h and k in equation (1), we obtain

(4 – 3)² + (1 – 4)² = r²

⇒ (1)² + (– 3)² = r²

⇒ 1 + 9 = r²

⇒ r² = 10

⇒

Thus, the equation of the required circle is

(x – 3)² + (y – 4)² =

x² – 6x + 9 + y² ­– 8y + 16 = 10

x² + y² – 6x – 8y + 15 = 0

Answer:

Let the equation of the required circle be (x – h)² + (y – k)² = r².

Since the circle passes through points (2, 3) and (–1, 1),

(2 – h)² + (3 – k)² = r² … (1)

(–1 – h)² + (1 – k)² = r² … (2)

Since the centre (h, k) of the circle lies on line x – 3y – 11 = 0,

h – 3k = 11 … (3)

From equations (1) and (2), we obtain

(2 – h)² + (3 – k)² = (–1 – h)² + (1 – k)²

⇒ 4 – 4h + h² + 9 – 6k + k² = 1 + 2h + h² + 1 – 2k + k²

⇒ 4 – 4h + 9 – 6k = 1 + 2h + 1 – 2k

⇒ 6h + 4k = 11 … (4)

On solving equations (3) and (4), we obtain.

On substituting the values of h and k in equation (1), we obtain

Thus, the equation of the required circle is

Answer:

Let the equation of the required circle be (x – h)² + (y – k)² = r².

Since the radius of the circle is 5 and its centre lies on the x-axis, k = 0 and r = 5.

Now, the equation of the circle becomes (x – h)² + y² = 25.

It is given that the circle passes through point (2, 3).

When h = –2, the equation of the circle becomes

(x + 2)² + y² = 25

x² + 4x + 4 + y² = 25

x² + y² + 4x – 21 = 0

When h = 6, the equation of the circle becomes

(x – 6)² + y² = 25

x² – 12x +36 + y² = 25

x² + y² – 12x + 11 = 0

Answer:

Let the equation of the required circle be (x – h)² + (y – k)² = r².

Since the circle passes through (0, 0),

(0 – h)² + (0 – k)² = r²

⇒ h² + k² = r²

The equation of the circle now becomes (x – h)² + (y – k)² = h² + k².

It is given that the circle makes intercepts a and b on the coordinate axes. This means that the circle passes through points (a, 0) and (0, b). Therefore,

(a – h)² + (0 – k)² = h² + k² … (1)

(0 – h)² + (b – k)² = h² + k² … (2)

From equation (1), we obtain

a² – 2ah + h² + k² = h² + k²

⇒ a² – 2ah = 0

⇒ a(a – 2h) = 0

⇒ a = 0 or (a – 2h) = 0

However, a ≠ 0; hence, (a – 2h) = 0 ⇒ h =.

From equation (2), we obtain

h² + b² – 2bk + k² = h² + k²

⇒ b² – 2bk = 0

⇒ b(b – 2k) = 0

⇒ b = 0 or(b – 2k) = 0

However, b ≠ 0; hence, (b – 2k) = 0 ⇒ k =.

Thus, the equation of the required circle is

Answer:

The centre of the circle is given as (h, k) = (2, 2).

Since the circle passes through point (4, 5), the radius (r) of the circle is the distance between the points (2, 2) and (4, 5).

Thus, the equation of the circle is

Answer:

The equation of the given circle is x² + y² = 25.

x² + y² = 25

⇒ (x – 0)² + (y – 0)² = 5², which is of the form (x – h)² + (y – k)² = r², where h = 0, k = 0, and r = 5.

∴Centre = (0, 0) and radius = 5

Distance between point (–2.5, 3.5) and centre (0, 0)

Since the distance between point (–2.5, 3.5) and centre (0, 0) of the circle is less than the radius of the circle, point (–2.5, 3.5) lies inside the circle.

Answer:

The given equation is y² = 12x.

Here, the coefficient of x is positive. Hence, the parabola opens towards the right.

On comparing this equation with y² = 4ax, we obtain

4a = 12 ⇒ a = 3

∴Coordinates of the focus = (a, 0) = (3, 0)

Since the given equation involves y², the axis of the parabola is the x-axis.

Equation of direcctrix, x = –a i.e., x = – 3 i.e., x + 3 = 0

Length of latus rectum = 4a = 4 × 3 = 12

Answer:

The given equation is x² = 6y.

Here, the coefficient of y is positive. Hence, the parabola opens upwards.

On comparing this equation with x² = 4ay, we obtain

∴Coordinates of the focus = (0, a) =

Since the given equation involves x², the axis of the parabola is the y-axis.

Equation of directrix,

Length of latus rectum = 4a = 6

Answer:

The given equation is y² = –8x.

Here, the coefficient of x is negative. Hence, the parabola opens towards the left.

On comparing this equation with y² = –4ax, we obtain

–4a = –8 ⇒ a = 2

∴Coordinates of the focus = (–a, 0) = (–2, 0)

Since the given equation involves y², the axis of the parabola is the x-axis.

Equation of directrix, x = a i.e., x = 2

Length of latus rectum = 4a = 8

Answer:

The given equation is x² = –16y.

Here, the coefficient of y is negative. Hence, the parabola opens downwards.

On comparing this equation with x² = – 4ay, we obtain

–4a = –16 ⇒ a = 4

∴Coordinates of the focus = (0, –a) = (0, –4)

Since the given equation involves x², the axis of the parabola is the y-axis.

Equation of directrix, y = a i.e., y = 4

Length of latus rectum = 4a = 16

Answer:

The given equation is y² = 10x.

Here, the coefficient of x is positive. Hence, the parabola opens towards the right.

On comparing this equation with y² = 4ax, we obtain

∴Coordinates of the focus = (a, 0)

Since the given equation involves y², the axis of the parabola is the x-axis.

Equation of directrix,

Length of latus rectum = 4a = 10

Answer:

The given equation is x² = –9y.

Here, the coefficient of y is negative. Hence, the parabola opens downwards.

On comparing this equation with x² = –4ay, we obtain

∴Coordinates of the focus =

Since the given equation involves x², the axis of the parabola is the y-axis.

Equation of directrix,

Length of latus rectum = 4a = 9

Answer:

Focus (6, 0); directrix, x = –6

Since the focus lies on the x-axis, the x-axis is the axis of the parabola.

Therefore, the equation of the parabola is either of the form y² = 4ax or

y² = – 4ax.

It is also seen that the directrix, x = –6 is to the left of the y-axis, while the focus (6, 0) is to the right of the y-axis. Hence, the parabola is of the form y² = 4ax.

Here, a = 6

Thus, the equation of the parabola is y² = 24x.

Answer:

Focus = (0, –3); directrix y = 3

Since the focus lies on the y-axis, the y-axis is the axis of the parabola.

Therefore, the equation of the parabola is either of the form x² = 4ay or

x² = – 4ay.

It is also seen that the directrix, y = 3 is above the x-axis, while the focus

(0, –3) is below the x-axis. Hence, the parabola is of the form x² = –4ay.

Here, a = 3

Thus, the equation of the parabola is x² = –12y.

Answer:

Vertex (0, 0); focus (3, 0)

Since the vertex of the parabola is (0, 0) and the focus lies on the positive x-axis, x-axis is the axis of the parabola, while the equation of the parabola is of the form y² = 4ax.

Since the focus is (3, 0), a = 3.

Thus, the equation of the parabola is y² = 4 × 3 × x, i.e., y² = 12x

Answer:

Vertex (0, 0) focus (–2, 0)

Since the vertex of the parabola is (0, 0) and the focus lies on the negative x-axis, x-axis is the axis of the parabola, while the equation of the parabola is of the form y² = –4ax.

Since the focus is (–2, 0), a = 2.

Thus, the equation of the parabola is y² = –4(2)x, i.e., y² = –8x

Answer:

Since the vertex is (0, 0) and the axis of the parabola is the x-axis, the equation of the parabola is either of the form y² = 4ax or y² = –4ax.

The parabola passes through point (2, 3), which lies in the first quadrant.

Therefore, the equation of the parabola is of the form y² = 4ax, while point

(2, 3) must satisfy the equation y² = 4ax.

Thus, the equation of the parabola is

Answer:

Since the vertex is (0, 0) and the parabola is symmetric about the y-axis, the equation of the parabola is either of the form x² = 4ay or x² = –4ay.

The parabola passes through point (5, 2), which lies in the first quadrant.

Therefore, the equation of the parabola is of the form x² = 4ay, while point

(5, 2) must satisfy the equation x² = 4ay.

Thus, the equation of the parabola is

Answer:

The given equation is.

Here, the denominator of is greater than the denominator of.

Therefore, the major axis is along the x-axis, while the minor axis is along the y-axis.

On comparing the given equation with, we obtain a = 6 and b = 4.

Therefore,

The coordinates of the foci are.

The coordinates of the vertices are (6, 0) and (–6, 0).

Length of major axis = 2a = 12

Length of minor axis = 2b = 8

Length of latus rectum

Answer:

The given equation is.

Here, the denominator of is greater than the denominator of.

Therefore, the major axis is along the y-axis, while the minor axis is along the x-axis.

On comparing the given equation with, we obtain b = 2 and a = 5.

Therefore,

The coordinates of the foci are.

The coordinates of the vertices are (0, 5) and (0, –5)

Length of major axis = 2a = 10

Length of minor axis = 2b = 4

Length of latus rectum

Answer:

The given equation is.

Here, the denominator of is greater than the denominator of.

Therefore, the major axis is along the x-axis, while the minor axis is along the y-axis.

On comparing the given equation with, we obtain a = 4 and b = 3.

Therefore,

The coordinates of the foci are.

The coordinates of the vertices are.

Length of major axis = 2a = 8

Length of minor axis = 2b = 6

Length of latus rectum

Answer:

The given equation is.

Here, the denominator of is greater than the denominator of.

Therefore, the major axis is along the y-axis, while the minor axis is along the x-axis.

On comparing the given equation with, we obtain b = 5 and a = 10.

Therefore,

The coordinates of the foci are.

The coordinates of the vertices are (0, ±10).

Length of major axis = 2a = 20

Length of minor axis = 2b = 10

Length of latus rectum

Answer:

The given equation is.

Here, the denominator of is greater than the denominator of.

Therefore, the major axis is along the x-axis, while the minor axis is along the y-axis.

On comparing the given equation with, we obtain a = 7 and b = 6.

Therefore,

The coordinates of the foci are.

The coordinates of the vertices are (± 7, 0).

Length of major axis = 2a = 14

Length of minor axis = 2b = 12

Length of latus rectum

Answer:

The given equation is.

Here, the denominator of is greater than the denominator of.

Therefore, the major axis is along the y-axis, while the minor axis is along the x-axis.

On comparing the given equation with, we obtain b = 10 and a = 20.

Therefore,

The coordinates of the foci are.

The coordinates of the vertices are (0, ±20)

Length of major axis = 2a = 40

Length of minor axis = 2b = 20

Length of latus rectum

Answer:

The given equation is 36x² + 4y² = 144.

It can be written as

Here, the denominator of is greater than the denominator of.

Therefore, the major axis is along the y-axis, while the minor axis is along the x-axis.

On comparing equation (1) with, we obtain b = 2 and a = 6.

Therefore,

The coordinates of the foci are.

The coordinates of the vertices are (0, ±6).

Length of major axis = 2a = 12

Length of minor axis = 2b = 4

Length of latus rectum

Answer:

The given equation is 16x² + y² = 16.

It can be written as

Here, the denominator of is greater than the denominator of.

Therefore, the major axis is along the y-axis, while the minor axis is along the x-axis.

On comparing equation (1) with, we obtain b = 1 and a = 4.

Therefore,

The coordinates of the foci are.

The coordinates of the vertices are (0, ±4).

Length of major axis = 2a = 8

Length of minor axis = 2b = 2

Length of latus rectum

Answer:

The given equation is 4x² + 9y² = 36.

It can be written as

Here, the denominator of is greater than the denominator of.

Therefore, the major axis is along the x-axis, while the minor axis is along the y-axis.

On comparing the given equation with, we obtain a = 3 and b = 2.

Therefore,

The coordinates of the foci are.

The coordinates of the vertices are (±3, 0).

Length of major axis = 2a = 6

Length of minor axis = 2b = 4

Length of latus rectum

Answer:

Vertices (±5, 0), foci (±4, 0)

Here, the vertices are on the x-axis.

Therefore, the equation of the ellipse will be of the form, where a is the semi-major axis.

Accordingly, a = 5 and c = 4.

It is known that.

Thus, the equation of the ellipse is.

Answer:

Vertices (0, ±13), foci (0, ±5)

Here, the vertices are on the y-axis.

Therefore, the equation of the ellipse will be of the form, where a is the semi-major axis.

Accordingly, a = 13 and c = 5.

It is known that.

Thus, the equation of the ellipse is.

Answer:

Vertices (±6, 0), foci (±4, 0)

Here, the vertices are on the x-axis.

Therefore, the equation of the ellipse will be of the form, where a is the semi-major axis.

Accordingly, a = 6, c = 4.

It is known that.

Thus, the equation of the ellipse is.

Answer:

Ends of major axis (±3, 0), ends of minor axis (0, ±2)

Here, the major axis is along the x-axis.

Therefore, the equation of the ellipse will be of the form, where a is the semi-major axis.

Accordingly, a = 3 and b = 2.

Thus, the equation of the ellipse is.

Answer:

Ends of major axis, ends of minor axis (±1, 0)

Here, the major axis is along the y-axis.

Therefore, the equation of the ellipse will be of the form, where a is the semi-major axis.

Accordingly, a = and b = 1.

Thus, the equation of the ellipse is.

Answer:

Length of major axis = 26; foci = (±5, 0).

Since the foci are on the x-axis, the major axis is along the x-axis.

Therefore, the equation of the ellipse will be of the form, where a is the semi-major axis.

Accordingly, 2a = 26 ⇒ a = 13 and c = 5.

It is known that.

Thus, the equation of the ellipse is.

Answer:

Length of minor axis = 16; foci = (0, ±6).

Since the foci are on the y-axis, the major axis is along the y-axis.

Therefore, the equation of the ellipse will be of the form, where a is the semi-major axis.

Accordingly, 2b = 16 ⇒ b = 8 and c = 6.

It is known that.

Thus, the equation of the ellipse is.

Answer:

Foci (±3, 0), a = 4

Since the foci are on the x-axis, the major axis is along the x-axis.

Therefore, the equation of the ellipse will be of the form, where a is the semi-major axis.

Accordingly, c = 3 and a = 4.

It is known that.

Thus, the equation of the ellipse is.

Answer:

It is given that b = 3, c = 4, centre at the origin; foci on the x axis.

Since the foci are on the x-axis, the major axis is along the x-axis.

Therefore, the equation of the ellipse will be of the form, where a is the semi-major axis.

Accordingly, b = 3, c = 4.

It is known that.

Thus, the equation of the ellipse is.

Answer:

Since the centre is at (0, 0) and the major axis is on the y-axis, the equation of the ellipse will be of the form

The ellipse passes through points (3, 2) and (1, 6). Hence,

On solving equations (2) and (3), we obtain b² = 10 and a² = 40.

Thus, the equation of the ellipse is.

Answer:

Since the major axis is on the x-axis, the equation of the ellipse will be of the form

The ellipse passes through points (4, 3) and (6, 2). Hence,

On solving equations (2) and (3), we obtain a² = 52 and b² = 13.

Thus, the equation of the ellipse is.

Answer:

The given equation is.

On comparing this equation with the standard equation of hyperbola i.e.,, we obtain a = 4 and b = 3.

We know that a² + b² = c².

Therefore,

The coordinates of the foci are (±5, 0).

The coordinates of the vertices are (±4, 0).

Length of latus rectum

Answer:

The given equation is.

On comparing this equation with the standard equation of hyperbola i.e.,, we obtain a = 3 and.

We know that a² + b² = c².

Therefore,

The coordinates of the foci are (0, ±6).

The coordinates of the vertices are (0, ±3).

Length of latus rectum

Answer:

The given equation is 9y² – 4x² = 36.

It can be written as

9y² – 4x² = 36

On comparing equation (1) with the standard equation of hyperbola i.e.,, we obtain a = 2 and b = 3.

We know that a² + b² = c².

Therefore,

The coordinates of the foci are.

The coordinates of the vertices are.

Length of latus rectum

Answer:

The given equation is 16x² – 9y² = 576.

It can be written as

16x² – 9y² = 576

On comparing equation (1) with the standard equation of hyperbola i.e.,, we obtain a = 6 and b = 8.

We know that a² + b² = c².

Therefore,

The coordinates of the foci are (±10, 0).

The coordinates of the vertices are (±6, 0).

Length of latus rectum

Answer:

The given equation is 5y² – 9x² = 36.

On comparing equation (1) with the standard equation of hyperbola i.e.,, we obtain a = and b = 2.

We know that a² + b² = c².

Therefore, the coordinates of the foci are.

The coordinates of the vertices are.

Length of latus rectum

Answer:

The given equation is 49y² – 16x² = 784.

It can be written as
49y² – 16x² = 784

On comparing equation (1) with the standard equation of hyperbola i.e.,, we obtain a = 4 and b = 7.

We know that a² + b² = c².

Therefore,

The coordinates of the foci are.

The coordinates of the vertices are (0, ±4).

Length of latus rectum

Answer:

Vertices (±2, 0), foci (±3, 0)

Here, the vertices are on the x-axis.

Therefore, the equation of the hyperbola is of the form.

Since the vertices are (±2, 0), a = 2.

Since the foci are (±3, 0), c = 3.

We know that a² + b² = c².

Thus, the equation of the hyperbola is.

Answer:

Vertices (0, ±5), foci (0, ±8)

Here, the vertices are on the y-axis.

Therefore, the equation of the hyperbola is of the form.

Since the vertices are (0, ±5), a = 5.

Since the foci are (0, ±8), c = 8.

We know that a² + b² = c².

Thus, the equation of the hyperbola is.

Answer:

Vertices (0, ±3), foci (0, ±5)

Here, the vertices are on the y-axis.

Therefore, the equation of the hyperbola is of the form.

Since the vertices are (0, ±3), a = 3.

Since the foci are (0, ±5), c = 5.

We know that a² + b² = c².

∴3² + b² = 5²

⇒ b² = 25 – 9 = 16

Thus, the equation of the hyperbola is.

Answer:

Foci (±5, 0), the transverse axis is of length 8.

Here, the foci are on the x-axis.

Therefore, the equation of the hyperbola is of the form.

Since the foci are (±5, 0), c = 5.

Since the length of the transverse axis is 8, 2a = 8 ⇒ a = 4.

We know that a² + b² = c².

∴4² + b² = 5²

⇒ b² = 25 – 16 = 9

Thus, the equation of the hyperbola is.

Answer:

Foci (0, ±13), the conjugate axis is of length 24.

Here, the foci are on the y-axis.

Therefore, the equation of the hyperbola is of the form.

Since the foci are (0, ±13), c = 13.

Since the length of the conjugate axis is 24, 2b = 24 ⇒ b = 12.

We know that a² + b² = c².

∴a² + 12² = 13²

⇒ a² = 169 – 144 = 25

Thus, the equation of the hyperbola is.

Answer:

Foci, the latus rectum is of length 8.

Here, the foci are on the x-axis.

Therefore, the equation of the hyperbola is of the form.

Since the foci are, c =.

Length of latus rectum = 8

We know that a² + b² = c².

∴a² + 4a = 45

⇒ a² + 4a – 45 = 0

⇒ a² + 9a – 5a – 45 = 0

⇒ (a + 9) (a – 5) = 0

⇒ a = –9, 5

Since a is non-negative, a = 5.

∴b² = 4a = 4 × 5 = 20

Thus, the equation of the hyperbola is.

Answer:

Foci (±4, 0), the latus rectum is of length 12.

Here, the foci are on the x-axis.

Therefore, the equation of the hyperbola is of the form.

Since the foci are (±4, 0), c = 4.

Length of latus rectum = 12

We know that a² + b² = c².

∴a² + 6a = 16

⇒ a² + 6a – 16 = 0

⇒ a² + 8a – 2a – 16 = 0

⇒ (a + 8) (a – 2) = 0

⇒ a = –8, 2

Since a is non-negative, a = 2.

∴b² = 6a = 6 × 2 = 12

Thus, the equation of the hyperbola is.

Answer:

Vertices (±7, 0),

Here, the vertices are on the x-axis.

Therefore, the equation of the hyperbola is of the form.

Since the vertices are (±7, 0), a = 7.

It is given that

We know that a² + b² = c².

Thus, the equation of the hyperbola is.

Answer:

Foci, passing through (2, 3)

Here, the foci are on the y-axis.

Therefore, the equation of the hyperbola is of the form.

Since the foci are, c =.

We know that a² + b² = c².

∴ a² + b² = 10

⇒ b² = 10 – a² … (1)

Since the hyperbola passes through point (2, 3),

From equations (1) and (2), we obtain

In hyperbola, c > a, i.e., c² > a²

∴ a² = 5

⇒ b² = 10 – a² = 10 – 5 = 5

Thus, the equation of the hyperbola is.

Answer:

The origin of the coordinate plane is taken at the vertex of the parabolic reflector in such a way that the axis of the reflector is along the positive x-axis.

This can be diagrammatically represented as

The equation of the parabola is of the form y² = 4ax (as it is opening to the right).

Since the parabola passes through point A (10, 5), 10² = 4a(5)

⇒ 100 = 20a

Therefore, the focus of the parabola is (a, 0) = (5, 0), which is the mid-point of the diameter.

Hence, the focus of the reflector is at the mid-point of the diameter.

Answer:

The origin of the coordinate plane is taken at the vertex of the arch in such a way that its vertical axis is along the negative y-axis.

This can be diagrammatically represented as

The equation of the parabola is of the form x² = $-$4ay (as it is opening downwards).

It can be clearly seen that the parabola passes through point $\left(\frac{5}{2},-10\right)$.

${\left(\frac{5}{2}\right)}^2=-4a\left(-10\right)$

$\Rightarrow 4a=\frac{25}{4\times 10}=\frac{5}{8}$

Therefore, the arch is in the form of a parabola whose equation is $x^2=-\frac{5}{8}y$.

When y = $-$2, $x^2=-\frac{5}{8}\left(-2\right)$

$\Rightarrow x^2=\frac{5}{4}$

$\Rightarrow x=\frac{\sqrt{5}}{2}$
​
$\therefore \mathrm{AB} = 2\times \frac{\sqrt{5}}{2}\mathrm{m} =\sqrt{5} \mathrm{m} = 2.23 \mathrm{m} (\mathrm{approx}.)$

Hence, when the arch is 2 m from the vertex of the parabola, its width is approximately 2.23 m.

Answer:

The vertex is at the lowest point of the cable. The origin of the coordinate plane is taken as the vertex of the parabola, while its vertical axis is taken along the positive y-axis. This can be diagrammatically represented as

Here, AB and OC are the longest and the shortest wires, respectively, attached to the cable.

DF is the supporting wire attached to the roadway, 18 m from the middle.

Here, AB = 30 m, OC = 6 m, and.

The equation of the parabola is of the form x² = 4ay (as it is opening upwards).

The coordinates of point A are (50, 30 – 6) = (50, 24).

Since A (50, 24) is a point on the parabola,

∴Equation of the parabola,or 6x² = 625y

The x-coordinate of point D is 18.

Hence, at x = 18,

∴DE = 3.11 m

DF = DE + EF = 3.11 m + 6 m = 9.11 m

Thus, the length of the supporting wire attached to the roadway 18 m from the middle is approximately 9.11 m.

Answer:

Since the height and width of the arc from the centre is 2 m and 8 m respectively, it is clear that the length of the major axis is 8 m, while the length of the semi-minor axis is 2 m.

The origin of the coordinate plane is taken as the centre of the ellipse, while the major axis is taken along the x-axis. Hence, the semi-ellipse can be diagrammatically represented as

The equation of the semi-ellipse will be of the form, where a is the semi-major axis

Accordingly, 2a = 8 ⇒ a = 4

b = 2

Therefore, the equation of the semi-ellipse is

Let A be a point on the major axis such that AB = 1.5 m.

Draw AC⊥ OB.

OA = (4 – 1.5) m = 2.5 m

The x-coordinate of point C is 2.5.

On substituting the value of x with 2.5 in equation (1), we obtain

∴AC = 1.56 m

Thus, the height of the arch at a point 1.5 m from one end is approximately 1.56 m.

Answer:

Let AB be the rod making an angle θ with OX and P (x, y) be the point on it such that AP = 3 cm.

Then, PB = AB – AP = (12 – 3) cm = 9 cm [AB = 12 cm]

From P, draw PQ⊥OY and PR⊥OX.

In ΔPBQ,

In ΔPRA,

Thus, the equation of the locus of point P on the rod is.

Answer:

The given parabola is x² = 12y.

On comparing this equation with x² = 4ay, we obtain 4a = 12 ⇒ a = 3

∴The coordinates of foci are S (0, a) = S (0, 3)

Let AB be the latus rectum of the given parabola.

The given parabola can be roughly drawn as

At y = 3, x² = 12 (3) ⇒ x² = 36 ⇒ x = ±6

∴The coordinates of A are (–6, 3), while the coordinates of B are (6, 3).

Therefore, the vertices of ΔOAB are O (0, 0), A (–6, 3), and B (6, 3).

Thus, the required area of the triangle is 18 unit².

Answer:

Let A and B be the positions of the two flag posts and P(x, y) be the position of the man. Accordingly, PA + PB = 10.

We know that if a point moves in a plane in such a way that the sum of its distances from two fixed points is constant, then the path is an ellipse and this constant value is equal to the length of the major axis of the ellipse.

Therefore, the path described by the man is an ellipse where the length of the major axis is 10 m, while points A and B are the foci.

Taking the origin of the coordinate plane as the centre of the ellipse, while taking the major axis along the x-axis, the ellipse can be diagrammatically represented as

The equation of the ellipse will be of the form, where a is the semi-major axis

Accordingly, 2a = 10 ⇒ a = 5

Distance between the foci (2c) = 8

⇒ c = 4

On using the relation, we obtain

Thus, the equation of the path traced by the man is.

Answer:

Let OAB be the equilateral triangle inscribed in parabola y² = 4ax.

Let AB intersect the x-axis at point C.

Let OC = k

From the equation of the given parabola, we have

∴The respective coordinates of points A and B are

AB = CA + CB =

Since OAB is an equilateral triangle, OA² = AB².

Thus, the side of the equilateral triangle inscribed in parabola y² = 4 ax is .