Find the $x$-intercept of the circle $x^2 + y^2 + 4x - 7y - 12 = 0$.

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

Let $P_1, P_2, P_3, P_4, P_5$ be five equally spaced points on the circumference of a circle of radius $1$,centred at $O$. Let $R$ be the set of points in the plane of the circle that are closer to $O$ than any of $P_1, P_2, P_3, P_4, P_5$. Then,$R$ is a

The angle subtended at the centre of a circle of radius $3 \text{ metres}$ by an arc of length $1 \text{ metre}$ is equal to

Let the normals at all the points on a given curve pass through a fixed point $(a, b)$. If the curve passes through $(3, -3)$ and $(4, -2\sqrt{2})$,and given that $a - 2\sqrt{2}b = 3$,then $(a^{2} + b^{2} + ab)$ is equal to ..... .

Suppose $C_1$ and $C_2$ are two circles having no common points,then