Conic Sections

Let a circle C touch the lines L₁ : 4x – 3y + K₁ = 0 and L₂ : 4x – 3y + K₂ = 0, K₁, K₂ $\in$  R. If a line passing through the centre of the circle C intersects L₁ at (-1,2) and L₂ at (3, -6), then the equation of the circle C is:

Let lx – 2y = µ be a tangent to the hyperbola $a^2{x}^2-y^2=b^2.$ Then ${\left(\frac{\lambda }{a}\right)}^2-{\left(\frac{\mu }{b}\right)}^2$ is equal to

Let x² + y² + Ax + By + C = 0 be a circle passing through (0, 6) and touching the parabola y = x² at (2, 4). Then A + C is equal to

Let P₁ be a parabola with vertex (3, 2) and focus (4, 4) and P₂ be its mirror image with respect to the line x + 2y = 6. Then the directrix P₂ is x + 2y =.

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: