JEE MAIN Mathematics Conic Sections MCQ 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 $O$ be the origin, and $P$ and $Q$ be two points on the rectangular hyperbola $xy=12$ such that the mid-point of the line segment $PQ$ is $\left(\frac{1}{2},-\frac{1}{2}\right)$. Then the area of the triangle $OPQ$ equals:

Let a circle pass through the origin and its centre be the point of intersection of two mutually perpendicular lines $x+(k-1)y+3=0$ and $2x+k^{2}y-4=0$. If the line $x-y+2=0$ intersects the circle at the points $A$ and $B$, then $(AB{)}^2$ 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: