The figure shows elliptical orbit of a planet m about the sun S. The shaded area SCD is twice the shaded area SAB. If be the time for the planet to move from C to D and is the time to move from A to B, then :

Velocity of efflux in Torricelli's theorem is given by , here h is the height of hole from the top surface, after that, motion of liquid can be treated as projectile motion. Liquid is filled in a vessel of square base up to a height of 2m as shown in figure (i). In figure (ii) the vessel is tilted from horizontal at What is the velocity of efflux in this case. Liquid does not spills out?

The number of values of c such that the straight line y = 4x + c touches the curve + y² = 1 is

If P(x, y), F₁=(3,0), F₂ = (– 3, 0)and16x² + 25 y² = 400, then P F₁ + P F₂ =

Let P be a variable point on the ellipse + = 1 with foci F₁ and F₂. If A is the area of the triangle PF₁ F₂, then the maximum value of A is-

Area of triangle with given vertices can be calculated using the matrix determimnant method.

The eccentricity of an ellipse, with its centre at the origin, is . If one of the directrices is x = 4, then the equation of the ellipse is-

standard equation of ellipse is
x²/a²+y²/b²=1

The foci of the ellipse + = 1 and the hyperbola – =  coincide. Then the value of b2 is-

focus of ellispe  = ae,0

The equation of an ellipse, whose major axis = 8 and eccentricity = 1/2, is

Arrangement of the following ellipses in ascending order of the radii of their director circles
P) 4x² + 9y² = 36
Q) 3x² + 4y² = 12
R) 9x² + 16y² = 144
S) x² + 2y² = 4

the radius of director circle is equal to the length of semi major axis.

If p and p' denote the lengths of the perpendicular from a focus and the centre of an ellipse with semi-major axis of length a, respectively, on a tangent to the ellipse and r denotes the focal distance of the point, then

the distances of the points from a line can be calculated by using the distance formula.

An ellipse and a hyperbola have the same centre “origin”, the same foci. The minor-axis of the one is the same as the conjugate axis of the other. If e₁, e₂ be their eccentricities respectively, then + is equal to

the conjugate axis of a hyperbola is the line through the center of the hyperbola and perpendicular to the line joining the focii.

If a circle of radius r is concentric with ellipse , then common tangent is inclined to the major axis at an angle-

the perpendicular distance from the tangent to the center of the circle is equal to the radius of the circle.

P is a point on the ellipse . The in-radius of ΔPSS’ (S and S’ are focii), where its area is maximum.

in radius is the radius of the circle inscribed inside the triangle.

Equation of one of the common tangent of y² = 4x and is equal to-

a common tangent is a line that is a tangent to more than one curves..

If F₁ and F₂ are the feet of the perpendiculars from the foci S₁ & S₂ of an ellipse on the tangent at any point P on the ellipse, then (S₁ F₁). (S₂ F₂) is equal to-

selecting a point on the major axis provides an ease of calculation.

The number of real tangents that can be drawn to the ellipse 3x² + 5y² = 32 and 25x² + 9y² = 450 passing through (3 , 5) is

if a point lies inside, no real tangents can be drawn from the point on the curve. on the curve, then only 1 tangent and if outside, then 2 tangents can be drawn.