JEE MAIN Mathematics Conic Sections Medium Previous Year Questions

Let the parabola $y=x^2+px+q$ passing through the point $(1,-1)$ be such that the distance between its vertex and the $x$-axis is minimum. Then the value of $p^2+q^2$ is:

Let an ellipse $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$, with $a<b$, pass through the point $(4,3)$ and have eccentricity $\frac{\sqrt{5}}{3}$. Then the length of its latus rectum is:

Let $A$ be the point $(3,0)$ and circles with variable diameter $AB$ touch the circle $x^2+y^2=36$ internally. Let the curve $C$ be the locus of the point $B$. If the eccentricity of $C$ is $e$, then $72e^2$ is equal to ____.

Let the focal chord $PQ$ of the parabola $y^2=4x$ make an angle of $60^{\circ }$ with the positive $x$-axis, where $P$ lies in the first quadrant. If the circle, whose one diameter is $PS$, $S$ being the focus of the parabola, touches the $y$-axis at the point $(0,\alpha )$, then $5{\alpha }^2$ is equal to: