The equation of the director circle of the circle ${x^2} + {y^2} = {a^2}$ is

For $t \in (0, 2\pi)$,if $ABC$ is an equilateral triangle with vertices $A(\sin t, -\cos t)$,$B(\cos t, \sin t)$,and $C(a, b)$ such that its orthocentre lies on a circle with centre $(1, 1/3)$,then $(a^2 - b^2)$ is equal to.

The locus of a point,which moves such that the sum of squares of its distances from the points $(0,0), (1,0), (0,1), (1,1)$ is $18$ units,is a circle of diameter $d$. Then $d^{2}$ is equal to ...... .

If $P(x, y)$ is a point such that the ratio of the squares of the lengths of the tangents from $P$ to the circles $x^2 + y^2 + 2x - 4y - 20 = 0$ and $x^2 + y^2 - 4x + 2y - 44 = 0$ is $2 : 3$,then the locus of $P$ is a circle with centre

From a point $A(0,3)$ on the circle $(x+2)^2+(y-3)^2=4$,a chord $AB$ is drawn and it is extended to a point $Q$ such that $AQ=2AB$. Then the locus of $Q$ is

The circle passing through the distinct points $(1, t)$,$(t, 1)$,and $(t, t)$ for all values of $t$ passes through the point: