JEE Main · 2026 · 22 Jan 2026 · Shift 1 · Circle

Let PQ and MN be two straight lines touching the circle $x^2+y^2-4x-6y-3=0$ at the points A and B respectively. Let O be the centre of the circle and $\angle \mathrm{AOB}=\pi /3$. Then the locus of the point of intersection of the lines PQ and MN is:

If P is a point on the circle $x^2+y^2=4$, Q is a point on the straight line $5x+y+2=0$ and $x-y+1=0$ is the perpendicular bisector of PQ, then 13 times the sum of abscissa of all such point P is ______.

Let a circle of radius 4 pass through the origin O , the points $\mathrm{A}(-\sqrt{3}\mathrm{a},0)$ and $\mathrm{B}(0,-\sqrt{2}\mathrm{b})$, where a and b are real parameters and $\mathrm{ab}\ne 0$. Then the locus of the centroid of $△\mathrm{OAB}$ is a cir …

Let $y=x$ be the equation of a chord of the circle ${\mathrm{C}}_1$ (in the closed half-plane $\mathrm{x}\ge 0$) of diameter 10 passing through the origin. Let ${\mathrm{C}}_2$ be another circle described on the given chord as its diameter. If the equation of the chord …

Let the circle $x^2+y^2=4$ intersect $x$-axis at the points $\mathrm{A}(\mathrm{a},0),\mathrm{a}>0$ and $\mathrm{B}(\mathrm{b},0)$. Let $\mathrm{P}(2\cos \alpha ,2\sin \alpha ),0<\alpha <\frac{\pi }{2}$ and $\mathrm{Q}(2\cos \beta ,2\sin \beta )$ be two points …

Let a circle C have its centre in the first quadrant, intersect the coordinate axes at exactly three points and cut off equal intercepts from the coordinate axes. If the length of the chord of C on the line $x+y=1$ is $\sqrt{14}$, then the square of the radius of C is ______.