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

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:

$A$ tangent is drawn to the parabola $y^2 = 4x$ at the point $P(t^2, 2t)$,where the abscissa $t^2$ lies in the interval $[1, 4]$. The maximum possible area of the triangle formed by the tangent at $P$,the ordinate of the point $P$,and the $x$-axis is equal to

The equation of the common tangent to the curves $y^2 = 8x$ and $xy = -1$ is:

Find the locus of a point which moves such that its distance from the point $(0, 0)$ is twice its distance from the $y$-axis.

Find the common tangent to the curves $x^2 + y^2 = 4$ and $2x^2 + y^2 = 2$.