Balbharati solutions for माठेमटिक्स अँड स्टॅटिस्टिक्स १ (आर्ट्स अँड सायन्स) [इंग्रजी] इयत्ता ११ महाराष्ट्र राज्य मंडळ chapter 7 - Conic Sections [Latest edition]

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:

5y² = 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:

y² = –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:

3x² = 8y

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:

x² = –8y

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:

3y² = –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 3y² = 16x, find the parameter of the point (3, – 4).

For the parabola 3y² = 16x, find the parameter of the point (27, –12).

Find the focal distance of a point on the parabola y² = 16x whose ordinate is 2 times the abscissa

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

y² = 12x whose parameter is `1/3`

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

2y² = 7x whose parameter is –2

For the parabola y² = 4x, find the coordinate of the point whose focal distance is 17

Find length of latus rectum of the parabola y² = 4ax passing through the point (2, –6)

Find the area of the triangle formed by the line joining the vertex of the parabola x² = 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 = x² – 2x + 3

Find the equation of tangent to the parabola y² = 12x from the point (2, 5)

Find the equation of tangent to the parabola y² = 36x from the point (2, 9)

If the tangent drawn from the point (–6, 9) to the parabola y² = kx are perpendicular to each other, find k

Two tangents to the parabola y² = 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 y² = 8(x + 2).

Find the equation of common tangent to the parabola y² = 4x and x² = 32y

Find the equation of the locus of a point, the tangents from which to the parabola y² = 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 x² + 16y = 0.

Show that the circle touches the directrix of the parabola.

Answer the following:

Find the

1. lengths of the principal axes
2. co-ordinates of the foci
3. equations of directrices
4. length of the latus rectum
5. distance between foci
6. distance between directrices of the ellipse:

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

Find the

1. lengths of the principal axes.
2. co-ordinates of the focii
3. equations of directrics
4. length of the latus rectum
5. distance between focii
6. distance between directrices of the ellipse:

3x² + 4y² = 12

Find the

1. lengths of the principal axes.
2. co-ordinates of the focii 
3. equations of directrics 
4. length of the latus rectum
5. distance between focii 
6. distance between directrices of the ellipse:

2x² + 6y² = 6

Find the 

1. lengths of the principal axes. 
2. co-ordinates of the focii 
3. equations of directrices 
4. length of the latus rectum
5. distance between focii 
6. distance between directrices of the ellipse:

3x² + 4y² = 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 3x² + 4y² = 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 9x² + 16y² = 144. Find the point of contact

Show that the line 8y + x = 17 touches the ellipse x² + 4y² = 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 9x² + 16y² = 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 4x² + 7y² = 28 from the point (3, –2).

Find the equation of the tangent to the ellipse 2x² + y² = 6 from the point (2, 1).

Find the equation of the tangent to the ellipse x² + 4y² = 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 5x² + 9y² = 45 which are ⊥ to the line 3x + 2y + y = 0.

Find the equation of the tangent to the ellipse x² + 4y² = 20, ⊥ to the line 4x + 3y = 7.

Find the equation of the locus of a point the tangents form which to the ellipse 3x² + 5y² = 15 are at right angles

Tangents are drawn through a point P to the ellipse 4x² + 5y² = 20 having inclinations θ₁ and θ₂ such that tan θ₁ + tan θ₂ = 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 θ₁ and θ₂. Find the equation of the locus of the point of intersection of the tangents at P and Q if θ₁ + θ₂ = `π/2`.

The eccentric angles of two points P and Q the ellipse 4x² + y² = 4 differ by `(2pi)/3`. Show that the locus of the point of intersection of the tangents at P and Q is the ellipse 4x² + y² = 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 3x² + 4y² = 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

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:

16x² – 9y² = 144

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:

21x² – 4y² = 84

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:

3x² – y² = 4

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:

x² – y² = 16

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:

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

`y^2/25 - x^2/144` = 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^2/100 - y^2/25` = + 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 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 x² – 3y² = 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:

3x² – y² = 4 at the point `(2, 2sqrt(2))`

Find the equation of the tangent to the hyperbola:

3x² – 4y² = 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:

9x² – 16y² = 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 x² – 4y² = 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 5x² – 4y² = 20 which are parallel to the line 3x + 2y + 12 = 0

Select the correct option from the given alternatives:

The line y = mx + 1 is a tangent to the parabola y² = 4x, if m is _______

Select the correct option from the given alternatives:

The length of latus rectum of the parabola x² – 4x – 8y + 12 = 0 is _________

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

Select the correct option from the given alternatives:

The coordinates of a point on the parabola y² = 8x whose focal distance is 4 are _______

Select the correct option from the given alternatives:

The endpoints of latus rectum of the parabola y² = 24x are _______

Select the correct option from the given alternatives:

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

Select the correct option from the given alternatives:

The area of the triangle formed by the line joining the vertex of the parabola x² = 12y to the endpoints of its latus rectum is _________

Select the correct option from the given alternatives:

If `"P"(pi/4)` is any point on he ellipse 9x² + 25y² = 225. S and S¹ are its foci then SP.S¹P =

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 _________

Select the correct option from the given alternatives:

If the parabola y² = 4ax passes through (3, 2) then the length of its latus rectum is ________

Select the correct option from the given alternatives

The eccentricity of rectangular hyperbola is

Select the correct option from the given alternatives:

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

Select the correct option from the given alternatives:

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

Select the correct option from the given alternatives:

If the line 4x − 3y + k = 0 touches the ellipse 5x² + 9y² = 45 then the value of k is

Select the correct option from the given alternatives:

The equation of the ellipse is 16x² + 25y² = 400. The equations of the tangents making an angle of 180° with the major axis are

Select the correct option from the given alternatives:

The equation of the tangent to the ellipse 4x² + 9y² = 36 which is perpendicular to the 3x + 4y = 17 is,

Select the correct option from the given alternatives:

Eccentricity of the hyperbola 16x² − 3y² − 32x − 12y − 44 = 0 is

Select the correct option from the given alternatives:

Centre of the ellipse 9x² + 5y² − 36x − 50y − 164 = 0 is at

Select the correct option from the given alternatives:

If the line 2x − y = 4 touches the hyperbola 4x² − 3y² = 24, the point of contact is

Select the correct option from the given alternatives:

The foci of hyperbola 4x² − 9y² − 36 = 0 are

Answer the following:

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

2y² = 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:

5x² = 24y

Answer the following:

Find the Cartesian coordinates of the point on the parabola y² = 12x whose parameter is 2

Answer the following:

Find the Cartesian coordinates of the point on the parabola y² = 12x whose parameter is −3

Answer the following:

Find the co-ordinates of a point of the parabola y² = 8x having focal distance 10

Answer the following:

Find the equation of the tangent to the parabola y² = 9x at the point (4, −6) on it

Answer the following:

Find the equation of the tangent to the parabola y² = 8x at t = 1 on it

Answer the following:

Find the equations of the tangents to the parabola y² = 9x through the point (4, 10).

Answer the following:

Show that the two tangents drawn to the parabola y² = 24x from the point (−6, 9) are at the right angle

Answer the following:

Find the equation of the tangent to the parabola y² = 8x which is parallel to the line 2x + 2y + 5 = 0. Find its point of contact

Answer the following:

A line touches the circle x² + y² = 2 and the parabola y² = 8x. Show that its equation is y = ± (x + 2).

Two tangents to the parabola y² = 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 y² = 8(x + 2).

Answer the following:

The slopes of the tangents drawn from P to the parabola y² = 4ax are m₁ and m₂, show that  m₁ − m₂ = k, where k is a constant.

Answer the following:

The slopes of the tangents drawn from P to the parabola y² = 4ax are m₁ and m₂, show that `("m"_1 /"m"_2)` = k, where k is a constant.

Answer the following:

The tangent at point P on the parabola y² = 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

1. lengths of the principal axes
2. co-ordinates of the foci
3. equations of directrices
4. length of the latus rectum
5. distance between foci
6. 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

16x² + 25y² = 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

x² − y² = 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 the length of major axis 10 and the distance between foci is 8

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

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 x² + 4y² = 100 at (8, 3)

Show that the line 8y + x = 17 touches the ellipse x² + 4y² = 17. Find the point of contact

Tangents are drawn through a point P to the ellipse 4x² + 5y² = 20 having inclinations θ₁ and θ₂ such that tan θ₁ + tan θ₂ = 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 7x² − 3y² = 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 4x² − 3y² = 24. Find the point of contact

Answer the following:

Find the equations of the tangents to the hyperbola 3x² − y² = 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 θ₁, θ₂, with the transverse axis. Find the locus of their point of intersection if tan θ₁ + tan θ₂ = k