Let P be the point of intersection of the common tangents to the parabola y2 = 12x and the hyperbola 8x2 – y2 = 8. If S and S' denote the foci of the hyperbola where S lies on the positive x-axis then P divides SS' in a ratio :

If the line ax + y = c, touches both the curves x2 + y2 = 1 and y2 = 4$$\sqrt{2}$$x , then |c| is equal to :

The area (in sq. units) of the smaller of the two circles that touch the parabola, y2 = 4x at the point (1, 2) and the x-axis is :-

If one end of a focal chord of the parabola, y2 = 16x is at (1, 4), then the length of this focal chord is :

The tangent to the parabola y2 = 4x at the point where it intersects the circle x2 + y2 = 5 in the first quadrant, passes through the point :

The shortest distance between the line y = x and the curve y2 = x – 2 is :

The equation of a tangent to the parabola, x2 = 8y, which makes an angle $$\theta$$ with the positive directions of x-axis, is :

The maximum area (in sq. units) of a rectangle having its base on the x-axis and its other two vertices on the parabola, y = 12 – x2 such that the rectangle lies inside the parabola, is :

Let P(4, –4) and Q(9, 6) be two points on the parabola, y2 = 4x and let x be any point on the arc POQ of this parabola, where O is the vertex of this parabola, such that the area of $$\Delta$$PXQ is maximum. Then this maximum area (in sq. units) is :

Axis of a parabola lies along x-axis. If its vertex and focus are at distances 2 and 4 respectively from the origin, on the positive x-axis then which of the following points does not lie on it?

Equation of a common tangent to the circle, x2 + y2 – 6x = 0 and the parabola, y2 = 4x is :

If $$\theta$$ denotes the acute angle between the curves, y = 10 – x2 and y = 2 + x2 at a point of their intersection, the |tan $$\theta$$| is equal to :

Let A(4, $$-$$ 4) and B(9, 6) be points on the parabola, y2 = 4x. Let C be chosen on the arc AOB of the parabola, where O is the origin, such that the area of $$\Delta$$ACB is maximum. Then, the area (in sq. units) of $$\Delta$$ACB, is :

If the parabolas y2 = 4b(x – c) and y2 = 8ax have a common normal, then which on of the following is a valid choice for the ordered triad (a, b, c)?

The length of the chord of the parabola x2 $$=$$ 4y having equation x – $$\sqrt{2}y+4\sqrt{2}=0$$ is -

If the area of the triangle whose one vertex is at the vertex of the parabola, y2 + 4(x – a2) = 0 and the othertwo vertices are the points of intersection of the parabola and y-axis, is 250 sq. units, then a value of 'a' is :

Let f be a differentiable function from R to R such that \left| {f\left( x \right) - f\left( y \right)} \right| \le 2{\left| {x - y} \right|^{{3 \over 2}}}, for all $$x,y\in$$ R. If $$f\left(0\right)=1$$ then $$\underset{0}{\overset{1}{\int }}{f}^2\left(x\right)dx$$ is equal to :

A value of $$\alpha$$ such that \int\limits_\alpha ^{\alpha + 1} {{{dx} \over {\left( {x + \alpha } \right)\left( {x + \alpha + 1} \right)}}} = {\log _e}\left( {{9 \over 8}} \right) is :

Let f : R $$\to$$ R be a continuously differentiable function such that f(2) = 6 and f'(2) = {1 \over {48}}. If $$\underset{6}{\overset{f\left(x\right)}{\int }}4t^{3}dt$$ = (x - 2)g(x), then $$\underset{x\to 2}{\lim }g\left(x\right)$$ is equal to :

If \int\limits_0^{{\pi \over 2}} {{{\cot x} \over {\cot x + \cos ecx}}} dx = m($$\pi$$ + n), then m.n is equal to