A triple star system consists of two stars, each of mass m, in the same circular orbit about central star with mass . The two outer stars always lie at opposite ends of a diameter of their common circular orbit. The radius of the circular orbit is r = m, and the orbital period of each star is s. [Take = 10 and G = ] The mass m of the outer stars is

A planet is revolving around the sun is an elliptical orbit as shown in figure. Select correct alternative(s)

If the law of gravitation be such that the force of attraction between two particles vary inversely as the power of their separation, then the graph of orbital velocity v₀ plotted against the distance r of a satellite from the earth's centre on a log-log scale is shown alongside. The slope of line will be-

A particle is projected from point A, that is at a distance 4R from the centre of the Earth, with speed in a direction making with the line joining the centre of the Earth and point A, as shown. Find the speed of particle (in m/s) if particle passes grazing the surface of the earth. Consider gravitational interaction only between these two.

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.