NCERT Exemplar solutions for Mathematics [English] Class 11 chapter 11 - Conic Sections [Latest edition]

Find the centre and radius of the circle x² + y² – 2x + 4y = 8

If the equation of the parabola is x² = – 8y, find coordinates of the focus, the equation of the directrix and length of latus rectum.

Given the ellipse with equation 9x² + 25y² = 225, find the major and minor axes, eccentricity, foci and vertices.

Find the equation of the ellipse with foci at (± 5, 0) and x = `36/5` as one of the directrices.

For the hyperbola 9x² – 16y² = 144, find the vertices, foci and eccentricity

Find the equation of the hyperbola with vertices at (0, ± 6) and e = `5/3`. Find its foci.

Find the equation of the circle which passes through the points (20, 3), (19, 8) and (2, –9). Find its centre and radius.

An equilateral triangle is inscribed in the parabola y² = 4ax whose one vertex is at the vertex of the parabola. Find the length of the side of the triangle.

Find the equation of the ellipse which passes through the point (–3, 1) and has eccentricity `sqrt(2)/5`, with x-axis as its major axis and centre at the origin.

Find the equation of the hyperbola whose vertices are (± 6, 0) and one of the directrices is x = 4.

The equation of the circle in the first quadrant touching each coordinate axis at a distance of one unit from the origin is ______.

The equation of the circle having centre (1, –2) and passing through the point of intersection of the lines 3x + y = 14 and 2x + 5y = 18 is ______.

The area of the triangle formed by the lines joining the vertex of the parabola x² = 12y to the ends of its latus rectum is ______.

The equations of the lines joining the vertex of the parabola y² = 6x to the points on it which have abscissa 24 are ______.

The equation of the ellipse whose centre is at the origin and the x-axis, the major axis, which passes through the points (–3, 1) and (2, –2) is ______.

The length of the transverse axis along x-axis with centre at origin of a hyperbola is 7 and it passes through the point (5, –2). The equation of the hyperbola is ______.

Circle on which the coordinates of any point are (2 + 4 cosθ, –1 + 4 sinθ) where θ is parameter is given by (x – 2)² + (y + 1)² = 16.

A bar of given length moves with its extremities on two fixed straight lines at right angles. Any point of the bar describes an ellipse.

The equation of the circle which passes through the point (4, 5) and has its centre at (2, 2) is ______.

A circle has radius 3 units and its centre lies on the line y = x – 1. If it passes through the point (7, 3), its equation is ______.

If the latus rectum of an ellipse with axis along x-axis and centre at origin is 10, distance between foci = length of minor axis, then the equation of the ellipse is ______.

The equation of the parabola whose focus is the point (2, 3) and directrix is the line x – 4y + 3 = 0 is ______.

The eccentricity of the hyperbola `x^2/a^2 - y^2/b^2` = 1 which passes through the points (3, 0) and `(3 sqrt(2), 2)` is ______.

Find the equation of the circle which touches the both axes in first quadrant and whose radius is a.

Show that the point (x, y) given by x = `(2at)/(1 + t^2)` and y = `(a(1 - t^2))/(1 - t^2)` lies on a circle for all real values of t such that –1 < t < 1 where a is any given real numbers.

If a circle passes through the point (0, 0) (a, 0), (0, b) then find the coordinates of its centre.

Find the equation of the circle which touches x-axis and whose centre is (1, 2).

If the lines 3x – 4y + 4 = 0 and 6x – 8y – 7 = 0 are tangents to a circle, then find the radius of the circle.

Find the equation of a circle which touches both the axes and the line 3x – 4y + 8 = 0 and lies in the third quadrant.

If one end of a diameter of the circle x² + y² – 4x – 6y + 11 = 0 is (3, 4), then find the coordinate of the other end of the diameter

Find the equation of the circle having (1, –2) as its centre and passing through 3x + y = 14, 2x + 5y = 18

If the line y = `sqrt(3)x + k` touches the circle x² + y² = 16, then find the value of k.

Find the equation of a circle concentric with the circle x² + y² – 6x + 12y + 15 = 0 and has double of its area.

If the latus rectum of an ellipse is equal to half of minor axis, then find its eccentricity.

Given the ellipse with equation 9x² + 25y² = 225, find the eccentricity and foci.

If the eccentricity of an ellipse is `5/8` and the distance between its foci is 10, then find latus rectum of the ellipse.

Find the equation of ellipse whose eccentricity is `2/3`, latus rectum is 5 and the centre is (0, 0).

Find the distance between the directrices of the ellipse `x^2/36 + y^2/20` = 1

Find the coordinates of a point on the parabola y² = 8x whose focal distance is 4.

Find the length of the line segment joining the vertex of the parabola y² = 4ax and a point on the parabola where the line segment makes an angle θ to the x-axis.

If the points (0, 4) and (0, 2) are respectively the vertex and focus of a parabola, then find the equation of the parabola.

If the line y = mx + 1 is tangent to the parabola y² = 4x then find the value of m.

If the distance between the foci of a hyperbola is 16 and its eccentricity is `sqrt(2)`, then obtain the equation of the hyperbola.

Find the eccentricity of the hyperbola 9y² – 4x² = 36.

Find the equation of the hyperbola with eccentricity `3/2` and foci at (± 2, 0).

If the lines 2x – 3y = 5 and 3x – 4y = 7 are the diameters of a circle of area 154 square units, then obtain the equation of the circle.

Find the equation of the circle which passes through the points (2, 3) and (4, 5) and the centre lies on the straight line y – 4x + 3 = 0.

Find the equation of a circle whose centre is (3, –1) and which cuts off a chord of length 6 units on the line 2x – 5y + 18 = 0.

Find the equation of a circle of radius 5 which is touching another circle x² + y² – 2x – 4y – 20 = 0 at (5, 5).

Find the equation of a circle passing through the point (7, 3) having radius 3 units and whose centre lies on the line y = x – 1.

Find the equation of the following parabolas:

Directrix x = 0, focus at (6, 0)

Find the equation of the following parabolas:

Vertex at (0, 4), focus at (0, 2)

Find the equation of the following parabolas:

Focus at (–1, –2), directrix x – 2y + 3 = 0

Find the equation of the set of all points the sum of whose distances from the points (3, 0) and (9, 0) is 12.

Find the equation of the set of all points whose distance from (0, 4) are `2/3` of their distance from the line y = 9.

Show that the set of all points such that the difference of their distances from (4, 0) and (– 4, 0) is always equal to 2 represent a hyperbola.

Find the equation of the hyperbola with vertices (± 5, 0), foci (± 7, 0)

Find the equation of the hyperbola with vertices (0, ± 7), e = `4/3`

Find the equation of the hyperbola with foci `(0, +- sqrt(10))`, passing through (2, 3)

The line x + 3y = 0 is a diameter of the circle x² + y² + 6x + 2y = 0.

The shortest distance from the point (2, –7) to the circle x² + y² – 14x – 10y – 151 = 0 is equal to 5.

If the line lx + my = 1 is a tangent to the circle x² + y² = a², then the point (l, m) lies on a circle.

The point (1, 2) lies inside the circle x² + y² – 2x + 6y + 1 = 0.

The line lx + my + n = 0 will touch the parabola y² = 4ax if ln = am².

If P is a point on the ellipse `x^2/16 + y^2/25` = 1 whose foci are S and S′, then PS + PS′ = 8.

The line 2x + 3y = 12 touches the ellipse `x^2/9 + y^2/4` = 2 at the point (3, 2).

The locus of the point of intersection of lines `sqrt(3)x - y - 4sqrt(3)k` = 0 and `sqrt(3)kx + ky - 4sqrt(3)` = 0 for different value of k is a hyperbola whose eccentricity is 2.

The equation of the circle having centre at (3, – 4) and touching the line 5x + 12y – 12 = 0 is ______.

The equation of the circle circumscribing the triangle whose sides are the lines y = x + 2, 3y = 4x, 2y = 3x is ______.

An ellipse is described by using an endless string which is passed over two pins. If the axes are 6 cm and 4 cm, the length of the string and distance between the pins are ______.

The equation of the ellipse having foci (0, 1), (0, –1) and minor axis of length 1 is ______. 

The equation of the parabola having focus at (–1, –2) and the directrix x – 2y + 3 = 0 is ______.

The equation of the hyperbola with vertices at (0, ± 6) and eccentricity `5/3` is ______ and its foci are ______.

The area of the circle centred at (1, 2) and passing through (4, 6) is ______.

Equation of a circle which passes through (3, 6) and touches the axes is ______.

Equation of the circle with centre on the y-axis and passing through the origin and the point (2, 3) is ______.

The equation of a circle with origin as centre and passing through the vertices of an equilateral triangle whose median is of length 3a is ______.

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

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

If the vertex of the parabola is the point (–3, 0) and the directrix is the line x + 5 = 0, then its equation is ______.

The equation of the ellipse whose focus is (1, –1), the directrix the line x – y – 3 = 0 and eccentricity `1/2` is ______.

The length of the latus rectum of the ellipse 3x² + y² = 12 is ______.

If e is the eccentricity of the ellipse `x^2/a^2 + y^2/b^2` = 1 (a < b), then ______.

The eccentricity of the hyperbola whose latus rectum is 8 and conjugate axis is equal to half of the distance between the foci is ______.

The distance between the foci of a hyperbola is 16 and its eccentricity is `sqrt(2)`. Its equation is ______.

Equation of the hyperbola with eccentricty `3/2` and foci at (± 2, 0) is ______.