The equation of the circle passing through (1, 5) and (4, 1) and touching y-axis is x² + y² − 5x − 6y + 9 + \(\lambda\)(4x + 3y − 19) = 0 where λ is equal to

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

The area of quadrilateral formed with foci of the hyperbolas \(\frac{x^{2}}{a^{2}}-\frac{y^{2}}{b^{2}}=1 \text { and } \frac{x^{2}}{a^{2}}-\frac{y^{2}}{b^{2}}=-1\)

If the normals of the parabola y² = 4x drawn at the end points of its latus rectum are tangents to the circle (x − 3)² + (y + 2)² = r² , then the value of r² is

If x + y = k is a normal to the parabola y² = 12x, then the value of k is

Consider an ellipse whose centre is of the origin and its major axis is along x-axis. If its eccentrcity is \(\frac { 3 }{ 5 } \) and the distance between its foci is 6, then the area of the quadrilateral inscribed in the ellipse with diagonals as major and minor axis of the ellipse is

Area of the greatest rectangle inscribed in the ellipse \(\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1\) is

If the coordinates at one end of a diameter of the circle x² + y² − 8x − 4y + c = 0 are (11, 2), the coordinates of the other end are

Equation of tangent at (-4, -4) on x² = -4y is _____________

y² - 2x - 2y + 5 = 0 is a _________

The distance between the foci of a hyperbola is 16 and e = \(\sqrt { 2 } \). Its equation is ____________

If the foci of the ellipse \(\frac { { x }^{ 2 } }{ 16 } +\frac { { y }^{ 2 } }{ { b }^{ 2 } } \) = 1 and the hyperbola \(\frac { { x }^{ 2 } }{ 144 } -\frac { { y }^{ 2 } }{ 81 } =\frac { 1 }{ 25 } \) coincide then b² is __________

The tangent at any point P on the ellipse \(\frac { { x }^{ 2 } }{ 6 } +\frac { { y }^{ 2 } }{ 3 } \) = 1 whose centre C meets the major axis at T and PN is the perpendicular to the major axis; The CN CT = ______________

The locus of the point of Intersection of perpendicular tangents to the hyperbela \(\frac { { x }^{ 2 } }{ 16 } +\frac { { y }^{ 2 } }{ 9 } \) = 1 is __________

The locus of the point of intersection of perpendicular tangents of the parabola y² = 4ax is

Find the general equation of a circle with centre (-3, -4) and radius 3 units.

Find the equation of the circle with centre (2, -1) and passing through the point (3, 6) in standard form.

Find centre and radius of the following circles.
x²+y²−x+2y−3 = 0

Find centre and radius of the following circles.
2x²+2y²−6x+4y+2 = 0

Find the equation of tangent to the circle x² +y² + 2x - 3y - 8 = 0 at (2, 3).

Find the equation of the parabola with vertex at the origin, passing through (2, -3) and symmetric about x-axis

If the line y = 3x + 1, touches the parabola y² = 4ax, find the length of the latus rectum?

For the ellipse x² + 3y² = a², find the length of major and minor axis.

Find the eccentricity of the hyperbola with foci on the x-axis if the length of its conjugate axis is \({ \left( \frac { 3 }{ 4 } \right) }^{ th }\) of the length of its tranverse axis.

Find the equation of the hyperbola whose vertices are (0, ±7) and e = \(\frac { 4 }{ 3 } \)

Find the equation of the circle described on the chord 3x + y + 5 = 0 of the circle x² + y² = 16 as diameter.

Find the centre and radius of the circle 3x² + (a + 1)y² + 6x − 9y + a + 4 = 0.

Find the equations of the tangent and normal to the circle x² + y² = 25 at P(-3, 4).

A circle of area 9π square units has two of its diameters along the lines x + y = 5 and x−y = 1. Find the equation of the circle.

If the equation 3x²+(3−p)xy+qy²−2px = 8pq represents a circle, find p and q. Also determine the centre and radius of the circle

Find the equation of the ellipse in each of the cases given below:
length of latus rectum 4, distance between foci 4 \( \sqrt{ 2}\) , centre (0, 0) and major axis as y - axis.

Find the equation of the hyperbola in each of the cases given below:
passing through (5, −2) and length of the transverse axis along x axis and of length 8 units.

Identify the type of the conic for the following equations:
(1) 16y² = −4x²+64
(2) x²+y² = −4x−y+4
(3) x²−2y = x+3
(4) 4x²−9y²−16x+18y−29 = 0

A search light has a parabolic reflector (has a cross-section that forms a ‘bowl’). The parabolic bowl is 40 cm wide from rim to rim and 30 cm deep. The bulb is located at the focus.
(1) What is the equation of the parabola used for reflector?
(2) How far from the vertex is the bulb to be placed so that the maximum distance covered?

For the hyperbola 3x² - 6y² = -18, find the length of transverse and conjugate axes and eccentricity.

Find the equation of the circle passing through the points (1, 1 ), (2, -1 ) and (3, 2) .

Find the foci, vertices and length of major and minor axis of the conic 4x² + 36y² + 40x − 288y + 532 = 0 

Find the centre, foci, and eccentricity of the hyperbola 11x² − 25y² −44x + 50y −256 = 0

A concrete bridge is designed as a parabolic arch. The road over bridge is 40 m long and the maximum height of the arch is 15 m. Write the equation of the parabolic arch.

Cross section of a Nuclear cooling tower is in the shape of a hyperbola with equation\(\frac { { x }^{ 2 } }{ { 30 }^{ 2 } } -\frac { { y }^{ 2 } }{ { 44 }^{ 2 } } =1\). The tower is 150 m tall and the distance from the top of the tower to the centre of the hyperbola is half the distance from the base of the tower to the centre of the hyperbola. Find the diameter of the top and base of the tower.
 

An arch is in the form of a parabola with its axis vertical. The arch is 10 m high and 5 m wide at the base. How wide is it 2 m from the vertex of the parabola?

A kho-kho player In a practice session while running realises that the sum of tne distances from the two kho-kho poles from him is always 8m. Find the equation of the path traced by him of the distance between the poles is 6m.