If the diameter of the circle $2(x^2 + y^2) + 3x + 4y - 1 = 0$ is $y = 2x + k$,then $k = \dots$

The equation of the circle with $(1,1)$ as centre and which cuts a chord of length $4 \sqrt{2}$ units on the line $x+y+1=0$ is

The maximum distance of the point $P(10, 7)$ from the circle $x^2 + y^2 - 4x - 2y - 20 = 0$ is:

Let $P$ and $Q$ be the inverse points with respect to the circle $S \equiv x^2+y^2-4x-6y+k=0$ and $C$ be the centre of the circle $S=0$ such that $CP \cdot CQ=4$. If $P=(1,2)$ and $Q=(a, b)$,then $2a=$

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

If $P(\frac{\pi}{3})$ and $Q(\frac{2\pi}{3})$ represent two points on the circle $x^2+y^2-4x+6y-12=0$ in parametric form,then the length of the chord $PQ$ is