Conic Sections

Let the hyperbola $H:\frac{x^2}{a^2}-y^2=1$ and the ellipse E : 3x² + 4y² = 12 be such that the length of latus rectum of H is equal to the length of latus rectum of E. If e_H and e_E are the eccentricities of H and E respectively, then the value of $12\left(e_H^2+e_E^2\right)$ is equal to

Let a circle C : ${\left(x-h\right)}^2+{\left(y-k\right)}^2=r^2,k>0,$ touch the axis (1,0). If the line x + y = 0 intersects the circle C at P and Q such that the length of the chord PQ is 2, then the value of h + k + r is equal to

The area (in sq. units) of region enclosed between the parabola y² = 2x and the line x + y = 4 is

Let the maximum area of the triangle that can be inscribed in the ellipse $\frac{x^2}{a^2}+\frac{y^2}{4}=1,a>2,$ having one of its vertices at one end of the major axis of the ellipse and one of its sides parallel to the y-axis, be $6\sqrt[]{3}$ . Then the eccentricity of the ellipse is

A particle is moving in the xy-plane along a curve C passing through the point (3, 3). The tangent to the curve C at the point P meets the x-axis at Q. If the y-axis bisects the segment PQ, then C is a parabola with

Let x = x(y) be the solution of the differential equation 2y $e^{x/y^{2}dx}+\left(y^2-4x{e}^{x/y^2}\right)dy=0$  such that x(1) = 0. Then, x(e) is equal to:

Let a triangle ABC be inscribed in the circle $x^2-\sqrt[]{2}\left(x+y\right)+y^2=0$ such that $\angle BAC=\frac{\pi }{2}.$ If the length of side AB is $\sqrt[]{2},$ then the area of the $\Delta ABC$ is equal to

Let P : y² = 4ax, a > 0 be a parabola with focus S. Let the tangents to the parabola P make an angle of  with the line y = 3x + 5 touch the parabola P at A and B. Then the value of a for which A, B and S are collinear is

If the solution of the differential equation $\frac{dy}{dx}+e^x\left(x^2-2\right)y=\left(x^2-2x\right)\left(x^2-2\right)e^{2x}$ $\frac{}{}{}^{}{}^{}{}^{}{}^{}{}^{}$ satisfies y(0) = 0, then the value of y(2) is……….

The area bounded by the curve y = $\left|x^2-9\right|$  and the  line y = 3 is :