A parabola with focus (3, 0) and directrix x = –3. Points P and Q lie on the parabola and their ordinates are in the ratio 3 : 1. The point of intersection of tangents drawn at points P and Q lies on the parabola

The set of all values of $a^2$ for which the line $x+y=0$ bisects two distinct chords drawn from a point $P \left(\frac{1+a}{2}, \frac{1-a}{2}\right)$ on the circle $2 x^2+2 y^2-(1+a) x-(1-a) y=0$, is equal to :

If the tangent at a point 𝑃 on the parabola $𝑦^2=3𝑥$ is parallel to the line $𝑥+2𝑦=1$ and the tangents at the points 𝑄 and 𝑅 on the ellipse $\frac{𝑥^2}{ 4} +\frac{𝑦^2}{ 1} =1$ are perpendicular to the line 𝑥−𝑦=2, then the area of the triangle PQR is :

Let the tangent to the parabola $y^2 = 12x$ at the point (3, α) be perpendicular to the line 2x + 2y = 3. Then the square of distance of the point (6, – 4) from the normal to the hyperbola $α^2 x^2 – 9y^2 = 9α^2$ at its point (α– 1, α + 2) is equal to__

Consider ellipse $ E_k : \frac{x^2}{k} + \frac{y^2}{k} = 1 $, for $ k = 1, 2, \dots, 20 $. Let $ C_k $ be the circle which touches the four chords joining the end points (one on the minor axis and another on the major axis) of the ellipse $ E_k $. If $ r_k $ is the radius of the circle $ C_k $, then the value of $ \sum_{k=1}^{20} r_k^2 $ is:

Let $ H_n : \frac{x^2}{1 + n} + \frac{y^2}{3 + n} = 1, n \in \mathbb{N} $. Let $ k $ be the smallest even value of $ n $ such that the eccentricity of $ H_n $ is a rational number. If $ l $ is the length of the latus rectum of $ H_k $, then 21 $ l $ is equal to:

The complex number $z=\frac{i-1}{\cos \frac{\pi}{3}+i \sin \frac{\pi}{3}}$ is equal to :

Let $z=1+i$ and $z_1=\frac{1+i \bar{z}}{\bar{z}(1-z)+\frac{1}{z}}$ Then $\frac{12}{\pi} \arg \left(z_1\right)$ is equal to________

Let the equation of the plane passing through the line $x-2 y-z-5=0=x+y+3 z-5$ and parallel to the line $x+y+2 z-7=0=2 x+3 y+z-2$ be $a x+b y+c z=65$ Then the distance of the point $(a, b, c)$ from the plane $2 x+2 y-z+16=0$ is _______

If $a ≠ ±b$ and are purely real, z ϵ complex number, $Re(az^{2}+bz)=a$ and $Re(bz^{2}+az)=b$ then number of value of z possible is