Circles Questions (554)

Let $P(a_1,b_1)$ and $Q(a_2,b_2)$ be two distinct points on a circle with centre $C(\sqrt{2},\sqrt{3})$. Let $O$ be the origin and $OC$ be perpendicular to both $CP$ and $CQ$. If the area of the triangle $OCP$ is $\dfrac{\sqrt{35}}{2}$, then $a_1^2+a_2^2+b_1^2+b_2^2$ is equal to ___.

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

If $(\alpha, \beta)$ is a point on a circle whose centre is on $x$-axis and has the coordinate $(\gamma, 0)$ which also touches the line $x + y = 0$ at $(2, -2)$, then:

Length of the tangents from the point \((1, 2)\) to the circles \(x^2 + y^2 + x + y - 4 = 0\) and \(3x^2 + 3y^2 - x - y - k = 0\) are in the ratio \(4:3\), then \(k\) is equal to