#### Chapters

## Chapter 7: Conic Sections

### Balbharati solutions for Mathematics and Statistics 1 (Arts and Science) 11th Standard Maharashtra State Board Chapter 7 Conic Sections Exercise 7.1 [Page 149]

Find co-ordinate of focus, equation of directrix, length of latus rectum and the co-ordinate of end points of latus rectum of the parabola:

5y^{2} = 24x

Find co-ordinate of focus, equation of directrix, length of latus rectum and the co-ordinate of end points of latus rectum of the parabola:

y^{2} = –20x

Find co-ordinate of focus, equation of directrix, length of latus rectum and the co-ordinate of end points of latus rectum of the parabola:

3x^{2} = 8y

x^{2} = –8y

3y^{2} = –16x

Find the equation of the parabola with vertex at the origin, axis along Y-axis and passing through the point (–10, –5).

Find the equation of the parabola with vertex at the origin, axis along X-axis and passing through the point (3, 4)

Find the equation of the parabola whose vertex is O(0, 0) and focus at (–7, 0).

Find the equation of the parabola with vertex at the origin, axis along X-axis and passing through the point (1, –6)

Find the equation of the parabola with vertex at the origin, axis along X-axis and passing through the point (2, 3)

For the parabola 3y^{2} = 16x, find the parameter of the point (3, – 4).

For the parabola 3y^{2} = 16x, find the parameter of the point (27, –12).

Find the focal distance of a point on the parabola y^{2} = 16x whose ordinate is 2 times the abscissa

Find coordinates of the point on the parabola. Also, find focal distance.

y^{2} = 12x whose parameter is `1/3`

Find coordinates of the point on the parabola. Also, find focal distance.

2y^{2} = 7x whose parameter is –2

For the parabola y^{2} = 4x, find the coordinate of the point whose focal distance is 17

Find length of latus rectum of the parabola y^{2} = 4ax passing through the point (2, –6)

Find the area of the triangle formed by the line joining the vertex of the parabola x^{2} = 12y to the end points of latus rectum.

If a parabolic reflector is 20 cm in diameter and 5 cm deep, find its focus.

Find coordinate of focus, vertex and equation of directrix and the axis of the parabola y = x^{2} – 2x + 3

Find the equation of tangent to the parabola y^{2} = 12x from the point (2, 5)

Find the equation of tangent to the parabola y^{2} = 36x from the point (2, 9)

If the tangent drawn from the point (–6, 9) to the parabola y^{2} = kx are perpendicular to each other, find k

Two tangents to the parabola y^{2} = 8x meet the tangents at the vertex in the point P and Q. If PQ = 4, prove that the equation of the locus of the point of intersection of two tangent is y^{2} = 8(x + 2).

Find the equation of common tangent to the parabola y^{2} = 4x and x^{2} = 32y

Find the equation of the locus of a point, the tangents from which to the parabola y^{2} = 18x are such that some of their slopes is –3

The tower of a bridge, hung in the form of a parabola have their tops 30 meters above the roadway and are 200 meters apart. If the cable is 5 meters above the roadway at the centre of the bridge, find the length of the vertical supporting cable 30 meters from the centre.

A circle whose centre is (4, –1) passes through the focus of the parabola x^{2} + 16y = 0.

Show that the circle touches the directrix of the parabola.

### Balbharati solutions for Mathematics and Statistics 1 (Arts and Science) 11th Standard Maharashtra State Board Chapter 7 Conic Sections Exercise 7.2 [Pages 163 - 164]

Answer the following:

Find the

- lengths of the principal axes
- co-ordinates of the foci
- equations of directrices
- length of the latus rectum
- distance between foci
- distance between directrices of the ellipse:

`x^2/25 + y^2/9` = 1

Find the

- lengths of the principal axes.
- co-ordinates of the focii
- equations of directrics
- length of the latus rectum
- distance between focii
- distance between directrices of the ellipse:

3x^{2} + 4y^{2} = 12

Find the

- lengths of the principal axes.
- co-ordinates of the focii
- equations of directrics
- length of the latus rectum
- distance between focii
- distance between directrices of the ellipse:

2x^{2} + 6y^{2} = 6

Find the

- lengths of the principal axes.
- co-ordinates of the focii
- equations of directrices
- length of the latus rectum
- distance between focii
- distance between directrices of the ellipse:

3x^{2} + 4y^{2} = 1

Find the equation of the ellipse in standard form if eccentricity = `3/8` and distance between its foci = 6

Find the equation of the ellipse in standard form if the length of major axis 10 and the distance between foci is 8

Find the equation of the ellipse in standard form if the distance between directrix is 18 and eccentricity is `1/3`.

Find the equation of the ellipse in standard form if the minor axis is 16 and eccentricity is `1/3`.

Find the equation of the ellipse in standard form if the distance between foci is 6 and the distance between directrix is `50/3`.

Find the equation of the ellipse in standard form if the latus rectum has length of 6 and foci are (±2, 0).

Find the equation of the ellipse in standard form if passing through the points (−3, 1) and (2, −2)

Find the equation of the ellipse in standard form if the dist. between its directrix is 10 and which passes through `(-sqrt(5), 2)`.

Find the equation of the ellipse in standard form if eccentricity is `2/3` and passes through `(2, −5/3)`.

Find the eccentricity of an ellipse, if the length of its latus rectum is one-third of its minor axis.

Find the eccentricity of an ellipse if the distance between its directrix is three times the distance between its foci

Show that the product of the lengths of the perpendicular segments drawn from the foci to any tangent line to the ellipse `x^2/25 + y^2/16` = 1 is equal to 16

A tangent having slope `–1/2` to the ellipse 3x^{2} + 4y^{2} = 12 intersects the X and Y axes in the points A and B respectively. If O is the origin, find the area of the triangle

Show that the line x – y = 5 is a tangent to the ellipse 9x^{2} + 16y^{2} = 144. Find the point of contact

Show that the line 8y + x = 17 touches the ellipse x^{2} + 4y^{2} = 17. Find the point of contact

Determine whether the line `x + 3ysqrt(2)` = 9 is a tangent to the ellipse `x^2/9 + y^2/4` = 1. If so, find the co-ordinates of the pt of contact

Find k, if the line 3x + 4y + k = 0 touches 9x^{2} + 16y^{2} = 144

Find the equation of the tangent to the ellipse `x^2/5 + y^2/4` = 1 passing through the point (2, –2)

Find the equation of the tangent to the ellipse 4x^{2} + 7y^{2} = 28 from the point (3, –2).

Find the equation of the tangent to the ellipse 2x^{2} + y^{2} = 6 from the point (2, 1).

Find the equation of the tangent to the ellipse x^{2} + 4y^{2} = 9 which are parallel to the line 2x + 3y – 5 = 0.

Find the equation of the tangent to the ellipse `x^2/25 + y^2/4` = 1 which are parallel to the line x + y + 1 = 0.

Find the equation of the tangent to the ellipse 5x^{2} + 9y^{2} = 45 which are ⊥ to the line 3x + 2y + y = 0.

Find the equation of the tangent to the ellipse x^{2} + 4y^{2} = 20, ⊥ to the line 4x + 3y = 7.

Find the equation of the locus of a point the tangents form which to the ellipse 3x^{2} + 5y^{2} = 15 are at right angles

Tangents are drawn through a point P to the ellipse 4x^{2} + 5y^{2} = 20 having inclinations θ_{1} and θ_{2} such that tan θ_{1} + tan θ_{2} = 2. Find the equation of the locus of P.

Show that the locus of the point of intersection of tangents at two points on an ellipse, whose eccentric angles differ by a constant, is an ellipse

P and Q are two points on the ellipse `x^2/"a"^2 + y^2/"b"^2` = 1 with eccentric angles θ_{1} and θ_{2}. Find the equation of the locus of the point of intersection of the tangents at P and Q if θ_{1} + θ_{2} = `π/2`.

The eccentric angles of two points P and Q the ellipse 4x^{2} + y^{2} = 4 differ by `(2pi)/3`. Show that the locus of the point of intersection of the tangents at P and Q is the ellipse 4x^{2} + y^{2} = 16

Find the equations of the tangents to the ellipse `x^2/16 + y^2/9` = 1, making equal intercepts on co-ordinate axes

A tangent having slope `–1/2` to the ellipse 3x^{2} + 4y^{2} = 12 intersects the X and Y axes in the points A and B respectively. If O is the origin, find the area of the triangle

### Balbharati solutions for Mathematics and Statistics 1 (Arts and Science) 11th Standard Maharashtra State Board Chapter 7 Conic Sections Exercise 7.3 [Pages 174 - 175]

Find the length of transverse axis, length of conjugate axis, the eccentricity, the co-ordinates of foci, equations of directrices and the length of latus rectum of the hyperbola:

`x^2/25 - y^2/16` = 1

Find the length of transverse axis, length of conjugate axis, the eccentricity, the co-ordinates of foci, equations of directrices and the length of latus rectum of the hyperbola:

`x^2/25 - y^2/16` = – 1

Find the length of transverse axis, length of conjugate axis, the eccentricity, the co-ordinates of foci, equations of directrices and the length of latus rectum of the hyperbola:

16x^{2} – 9y^{2} = 144

21x^{2} – 4y^{2} = 84

3x^{2} – y^{2} = 4

x^{2} – y^{2} = 16

`y^2/25 - x^2/9` = 1

`y^2/25 - x^2/144` = 1

`x^2/100 - y^2/25` = + 1

x = 2 sec θ, y = `2sqrt(3) tan theta`

Find the equation of the hyperbola with centre at the origin, length of conjugate axis 10 and one of the foci (–7, 0).

Find the eccentricity of the hyperbola, which is conjugate to the hyperbola x^{2} – 3y^{2} = 3

If e and e' are the eccentricities of a hyperbola and its conjugate hyperbola respectively, prove that `1/"e"^2 + 1/("e""'")^2` = 1

Find the equation of the hyperbola referred to its principal axes:

whose distance between foci is 10 and eccentricity `5/2`

Find the equation of the hyperbola referred to its principal axes:

whose distance between foci is 10 and length of conjugate axis 6

Find the equation of the hyperbola referred to its principal axes:

whose distance between directrices is `8/3` and eccentricity is `3/2`

Find the equation of the hyperbola referred to its principal axes:

whose length of conjugate axis = 12 and passing through (1, – 2)

Find the equation of the hyperbola referred to its principal axes:

which passes through the points (6, 9) and (3, 0)

Find the equation of the hyperbola referred to its principal axes:

whose vertices are (± 7, 0) and end points of conjugate axis are (0, ±3)

Find the equation of the hyperbola referred to its principal axes:

whose foci are at (±2, 0) and eccentricity `3/2`

Find the equation of the hyperbola referred to its principal axes:

whose length of transverse and conjugate axis are 6 and 9 respectively

Find the equation of the hyperbola referred to its principal axes:

whose length of transverse axis is 8 and distance between foci is 10

Find the equation of the tangent to the hyperbola:

3x^{2} – y^{2} = 4 at the point `(2, 2sqrt(2))`

Find the equation of the tangent to the hyperbola:

3x^{2} – 4y^{2} = 12 at the point (4, 3)

Find the equation of the tangent to the hyperbola:

`x^2/144 - y^2/25` = 1 at the point whose eccentric angle is `pi/3`

Find the equation of the tangent to the hyperbola:

`x^2/16 - y^2/9` = 1 at the point in a first quadratures whose ordinate is 3

Find the equation of the tangent to the hyperbola:

9x^{2} – 16y^{2} = 144 at the point L of latus rectum in the first quadrant

Show that the line 3x – 4y + 10 = 0 is tangent till the hyperbola x^{2} – 4y^{2} = 20. Also find the point of contact

If the 3x – 4y = k touches the hyperbola `x^2/5 - (4y^2)/5` = 1 then find the value of k

Find the equations of the tangents to the hyperbola `x^2/25 - y^2/9` = 1 making equal intercepts on the co-ordinate axes

Find the equations of the tangents to the hyperbola 5x^{2} – 4y^{2} = 20 which are parallel to the line 3x + 2y + 12 = 0

### Balbharati solutions for Mathematics and Statistics 1 (Arts and Science) 11th Standard Maharashtra State Board Chapter 7 Conic Sections Miscellaneous Exercise 7 [Pages 176 - 178]

Select the correct option from the given alternatives:

The line y = mx + 1 is a tangent to the parabola y^{2} = 4x, if m is _______

1

2

3

4

Select the correct option from the given alternatives:

The length of latus rectum of the parabola x^{2} – 4x – 8y + 12 = 0 is _________

4

6

8

10

Select the correct option from the given alternatives:

If the focus of the parabola is (0, –3) its directrix is y = 3 then its equation is

x

^{2}= – 12yx

^{2}= 12yy

^{2}= 12xy

^{2}= −12x

Select the correct option from the given alternatives:

The coordinates of a point on the parabola y^{2} = 8x whose focal distance is 4 are _______

`(1/2, ±2)`

`(1, ±2sqrt(2))`

(2, ± 4)

none of these

Select the correct option from the given alternatives:

The endpoints of latus rectum of the parabola y^{2} = 24x are _______

(6, ±12)

(12, ±6)

(6, ±6)

none of these

Select the correct option from the given alternatives:

Equation of the parabola with vertex at the origin and directrix x + 8 = 0 is __________

y

^{2}= 8xy

^{2}= 32xy

^{2}= 16xx

^{2}= 32y

Select the correct option from the given alternatives:

The area of the triangle formed by the line joining the vertex of the parabola x^{2} = 12y to the endpoints of its latus rectum is _________

22 sq.units

20 sq.units

18 sq.units

14 sq.units

Select the correct option from the given alternatives:

If `"P"(pi/4)` is any point on he ellipse 9x^{2} + 25y^{2} = 225. S and S^{1} are its foci then SP.S^{1}P =

13

14

17

19

Select the correct option from the given alternatives:

The equation of the parabola having (2, 4) and (2, –4) as endpoints of its latus rectum is _________

y

^{2}= 4xy

^{2}= 8xy

^{2}= –16xx

^{2}= 8y

Select the correct option from the given alternatives:

If the parabola y^{2} = 4ax passes through (3, 2) then the length of its latus rectum is ________

`2/3`

`4/3`

`1/3`

4

Select the correct option from the given alternatives

The eccentricity of rectangular hyperbola is

`1/2`

`1/(2 1/2)`

`2 1/2`

`1/(3 1/2)`

Select the correct option from the given alternatives:

The equation of the ellipse having foci (+4, 0) and eccentricity `1/3` is

9x

^{2}+ 16y^{2}= 144144x

^{2}+ 9y^{2}= 1296128x

^{2}+ 144y^{2}= 18432144x

^{2}+ 128y^{2}= 18432

Select the correct option from the given alternatives:

The equation of the ellipse having eccentricity `sqrt(3)/2` and passing through (− 8, 3) is

4x

^{2}+ y^{2}= 4x

^{2}+ 4y^{2}= 1004x

^{2}+ y^{2}= 100x

^{2}+ 4y^{2}= 4

Select the correct option from the given alternatives:

If the line 4x − 3y + k = 0 touches the ellipse 5x^{2} + 9y^{2} = 45 then the value of k is

+ 21

`± 3sqrt(21)`

+ 3

+ 3(21)

Select the correct option from the given alternatives:

The equation of the ellipse is 16x^{2} + 25y^{2} = 400. The equations of the tangents making an angle of 180° with the major axis are

x = 4

y = ± 4

x = – 4

x = ± 5

Select the correct option from the given alternatives:

The equation of the tangent to the ellipse 4x^{2} + 9y^{2} = 36 which is perpendicular to the 3x + 4y = 17 is,

y = 4x + 6

3y + 4x = 6

3y = `4x + 6sqrt(5)`

3y = x + 25

Select the correct option from the given alternatives:

Eccentricity of the hyperbola 16x^{2} − 3y^{2} − 32x − 12y − 44 = 0 is

`sqrt(17/3)`

`sqrt(19/3)`

`sqrt(19)/3`

`sqrt(17)/3`

Select the correct option from the given alternatives:

Centre of the ellipse 9x^{2} + 5y^{2} − 36x − 50y − 164 = 0 is at

(2, 5)

(1, −2)

(−2, 1)

(0, 0)

Select the correct option from the given alternatives:

If the line 2x − y = 4 touches the hyperbola 4x^{2} − 3y^{2} = 24, the point of contact is

(1, 2)

(2, 3)

(3, 2)

(−2, −3)

Select the correct option from the given alternatives:

The foci of hyperbola 4x^{2} − 9y^{2} − 36 = 0 are

`(± sqrt(13), 0)`

`(± sqrt(11), 0)`

`(± sqrt(12), 0)`

` (0,± sqrt(12))`

Answer the following:

For the following parabola, find focus, equation of the directrix, length of the latus rectum, and ends of the latus rectum:

2y^{2 }= 17x

Answer the following:

For the following parabola, find focus, equation of the directrix, length of the latus rectum, and ends of the latus rectum:

5x^{2 }= 24y

Answer the following:

Find the Cartesian coordinates of the point on the parabola y^{2} = 12x whose parameter is 2

Answer the following:

Find the Cartesian coordinates of the point on the parabola y^{2} = 12x whose parameter is −3

Answer the following:

Find the co-ordinates of a point of the parabola y^{2} = 8x having focal distance 10

Answer the following:

Find the equation of the tangent to the parabola y^{2} = 9x at the point (4, −6) on it

Answer the following:

Find the equation of the tangent to the parabola y^{2} = 8x at t = 1 on it

Answer the following:

Find the equations of the tangents to the parabola y^{2} = 9x through the point (4, 10).

Answer the following:

Show that the two tangents drawn to the parabola y^{2} = 24x from the point (−6, 9) are at the right angle

Answer the following:

Find the equation of the tangent to the parabola y^{2} = 8x which is parallel to the line 2x + 2y + 5 = 0. Find its point of contact

Answer the following:

A line touches the circle x^{2} + y^{2} = 2 and the parabola y^{2} = 8x. Show that its equation is y = ± (x + 2).

Two tangents to the parabola y^{2} = 8x meet the tangents at the vertex in the point P and Q. If PQ = 4, prove that the equation of the locus of the point of intersection of two tangent is y^{2} = 8(x + 2).

Answer the following:

The slopes of the tangents drawn from P to the parabola y^{2} = 4ax are m_{1} and m_{2}, show that m_{1} − m_{2} = k, where k is a constant.

Answer the following:

The slopes of the tangents drawn from P to the parabola y^{2} = 4ax are m_{1} and m_{2}, show that `("m"_1 /"m"_2)` = k, where k is a constant.

Answer the following:

The tangent at point P on the parabola y^{2} = 4ax meets the y-axis in Q. If S is the focus, show that SP subtends a right angle at Q

Answer the following:

Find the

- lengths of the principal axes
- co-ordinates of the foci
- equations of directrices
- length of the latus rectum
- distance between foci
- distance between directrices of the ellipse:

`x^2/25 + y^2/9` = 1

Answer the following:

Find the

(i) lengths of the principal axes

(ii) co-ordinates of the foci

(iii) equations of directrices

(iv) length of the latus rectum

(v) Distance between foci

(vi) distance between directrices of the curve

16x^{2} + 25y^{2} = 400

Answer the following:

Find the

(i) lengths of the principal axes

(ii) co-ordinates of the foci

(iii) equations of directrices

(iv) length of the latus rectum

(v) Distance between foci

(vi) distance between directrices of the curve

`x^2/144 - y^2/25` = 1

Answer the following:

Find the

(i) lengths of the principal axes

(ii) co-ordinates of the foci

(iii) equations of directrices

(iv) length of the latus rectum

(v) Distance between foci

(vi) distance between directrices of the curve

x^{2} − y^{2} = 16

Find the equation of the ellipse in standard form if eccentricity = `3/8` and distance between its foci = 6

Find the equation of the ellipse in standard form if passing through the points (−3, 1) and (2, −2)

Find the equation of the ellipse in standard form if the length of major axis 10 and the distance between foci is 8

Find the eccentricity of an ellipse if the distance between its directrix is three times the distance between its foci

Answer the following:

For the hyperbola `x^2/100−y^2/25` = 1, prove that SA. S'A = 25, where S and S' are the foci and A is the vertex

Find the equation of the tangent to the ellipse `x^2/5 + y^2/4` = 1 passing through the point (2, –2)

Answer the following:

Find the equation of the tangent to the ellipse x^{2} + 4y^{2} = 100 at (8, 3)

Show that the line 8y + x = 17 touches the ellipse x^{2} + 4y^{2} = 17. Find the point of contact

Tangents are drawn through a point P to the ellipse 4x^{2} + 5y^{2} = 20 having inclinations θ_{1} and θ_{2} such that tan θ_{1} + tan θ_{2} = 2. Find the equation of the locus of P.

Show that the product of the lengths of the perpendicular segments drawn from the foci to any tangent line to the ellipse `x^2/25 + y^2/16` = 1 is equal to 16

Answer the following:

Find the equation of the hyperbola in the standard form if Length of conjugate axis is 5 and distance between foci is 13.

Answer the following:

Find the equation of the hyperbola in the standard form if eccentricity is `3/2` and distance between foci is 12.

Answer the following:

Find the equation of the hyperbola in the standard form if length of the conjugate axis is 3 and distance between the foci is 5.

Answer the following:

Find the equation of the tangent to the hyperbola 7x^{2} − 3y^{2} = 51 at (−3, −2)

Answer the following:

Find the equation of the tangent to the hyperbola x = 3 secθ, y = 5 tanθ at θ = `pi/3`

Answer the following:

Find the equation of the tangent to the hyperbola `x^2/25 − y^2/16` = 1 at P(30°)

Answer the following:

Show that the line 2x − y = 4 touches the hyperbola 4x^{2} − 3y^{2} = 24. Find the point of contact

Answer the following:

Find the equations of the tangents to the hyperbola 3x^{2} − y^{2} = 48 which are perpendicular to the line x + 2y − 7 = 0

Answer the following:

Two tangents to the hyperbola `x^2/"a"^2 - y^2/"b"^2` = 1 make angles θ_{1}, θ_{2}, with the transverse axis. Find the locus of their point of intersection if tan θ_{1} + tan θ_{2} = k

## Chapter 7: Conic Sections

## Balbharati solutions for Mathematics and Statistics 1 (Arts and Science) 11th Standard Maharashtra State Board chapter 7 - Conic Sections

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