Circles JEE Main PYQs, 2020 to 2024

Let a circle passing through $(2,0)$ have its centre at the point $(\mathrm{h}, \mathrm{k})$. Let $(x_{\mathrm{c}}, y_{\mathrm{c}})$ be the point of intersection of the lines $3 x+5 y=1$ and $(2+\mathrm{c}) x+5 \mathrm{c}^2 y=1$. If $\mathrm{h}=\lim \limits_{\mathrm{c} \rightarrow 1} x_{\mathrm{c}}$ and $\mathrm{k}=\lim \limits_{\mathrm{c} \rightarrow 1} y_{\mathrm{c}}$, then the equation of the circle is :

Let the centre of a circle, passing through the points $(0,0),(1,0)$ and touching the circle $x^2+y^2=9$, be $(h, k)$. Then for all possible values of the coordinates of the centre $(h, k), 4\left(h^2+k^2\right)$ is equal to __________.

Let the circle $C_1: x^2+y^2-2(x+y)+1=0$ and $\mathrm{C_2}$ be a circle having centre at $(-1,0)$ and radius 2 . If the line of the common chord of $\mathrm{C}_1$ and $\mathrm{C}_2$ intersects the $\mathrm{y}$-axis at the point $\mathrm{P}$, then the square of the distance of P from the centre of $\mathrm{C_1}$ is:

A circle with centre (2, 3) and radius 4 intersects the line $x+y=3$ at the points P and Q. If the tangents at P and Q intersect at the point $S(\alpha,\beta)$, then $4\alpha-7\beta$ is equal to ___________.