Let $P$ be a moving point on the circle $x^2 + y^2 - 6x - 8y + 21 = 0$. Then, the maximum distance of $P$ from the vertex of the parabola $x^2 + 6x + y + 13 = 0$ is equal to:

Suppose $a, b$ are real numbers such that $ab \neq 0$. Which of the following four figures represents the curve $(y-ax-b)(bx^2+ay^2-ab)=0$?

How many parabolas can be drawn if the endpoints of the latus rectum are given?

If the curves $2x^2 + ky^2 = 30$ and $3y^2 = 28x$ cut each other orthogonally,then $k=$

Let $P$ be the point of intersection of the common tangents to the parabola $y^2 = 12x$ and the hyperbola $8x^2 - y^2 = 8$. If $S$ and $S'$ denote the foci of the hyperbola where $S$ lies on the positive $x$-axis,then $P$ divides $SS'$ in the ratio:

Find the equation of the ellipse whose focus is $(-1, 1)$,eccentricity is $1/2$,and directrix is $x - y + 3 = 0$.