$A$ circle passes through the point $(0, 1)$ and touches the parabola $y = x^2$ at the point $(1, 1)$. The centre of the circle is...

If the product of the lengths of the perpendiculars drawn from the ends of a diameter of the circle $x^2+y^2=4$ onto the line $x+y+1=0$ is maximum,then the two ends of that diameter are

The equation of the circumcircle of the triangle formed by the lines $y + \sqrt{3}x = 6$,$y - \sqrt{3}x = 6$,and $y = 0$ is:

The sum of the minimum and maximum distances of the point $(4,-3)$ to the circle $x^2+y^2+4x-10y-7=0$ is

Suppose $Q$ is a point on the circle with centre $P$ and radius $1$,as shown in the figure. $R$ is a point outside the circle such that $QR = 1$ and $\angle QRP = 2^{\circ}$. Let $S$ be the point where the segment $RP$ intersects the given circle. Then,the measure of $\angle RQS$ equals $......^{\circ}$.

Let $R$ be the region of the disc $x^2+y^2 \leq 1$ in the first quadrant. Then,the area of the largest possible circle contained in $R$ is