JEE Main 2025 (Online) 4th April Evening Shift | Parabola Question 4 | Mathematics

The axis of a parabola is the line $y=x$ and its vertex and focus are in the first quadrant at distances $\sqrt{2}$ and $2 \sqrt{2}$ units from the origin, respectively. If the point $(1, k)$ lies on the parabola, then a possible value of k is :

JEE Main 2025 (Online) 3rd April Morning Shift

MCQ (Single Correct Answer)

-1

The radius of the smallest circle which touches the parabolas $y=x^2+2$ and $x=y^2+2$ is

$\frac{7 \sqrt{2}}{16}$

$\frac{7 \sqrt{2}}{8}$

$\frac{7 \sqrt{2}}{2}$

$\frac{7 \sqrt{2}}{4}$

JEE Main 2025 (Online) 2nd April Evening Shift

MCQ (Single Correct Answer)

-1

Let the point P of the focal chord PQ of the parabola $y^2=16 x$ be $(1,-4)$. If the focus of the parabola divides the chord $P Q$ in the ratio $m: n, \operatorname{gcd}(m, n)=1$, then $m^2+n^2$ is equal to :

JEE Main 2025 (Online) 2nd April Morning Shift

MCQ (Single Correct Answer)

-1

Let the focal chord PQ of the parabola $y^2=4 x$ 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: