The locus of the centre of a circle which passes through the point $(a, 0)$ and touches the line $x + 1 = 0$ is

Two perpendicular tangents to the circle $x^2 + y^2 = a^2$ meet at point $P$. The equation of the locus of $P$ is:

If the ratio of the distances of a variable point $P$ from the point $(1, 1)$ and the line $x-y+2=0$ is $1: \sqrt{2}$,then the equation of the locus of $P$ is

$P$ is a variable point such that the distance of $P$ from $A(4,0)$ is twice the distance of $P$ from $B(-4,0)$. If the line $3y - 3x - 20 = 0$ intersects the locus of $P$ at the points $C$ and $D$,then the distance between $C$ and $D$ is:

The tangent at any point on the curve $x^2 + y^2 = r^2$ meets the coordinate axes at $A$ and $B$. If lines are drawn through $A$ and $B$ parallel to the coordinate axes to intersect at $P$,find the locus of $P$.

$A$ company situated at $(2,0)$ in the $XY$-plane charges $RS. 2$ per $km$ for delivery. $A$ second company at $(0,3)$ charges $RS. 3$ per $km$ for delivery. The region of the plane where it is cheaper to use the first company is