JEE Main · 2023 · 29 Jan 2023 · 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 the set of all values of r, for which the circles $(x+1{)}^2+(y+4{)}^2=r^2$ and $x^2+y^2-4x-2y-4=0$ intersect at two distinct points be the interval ($\alpha$, $\beta$). Then $\alpha \beta$ is equal to

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 …