If all chords of the curve $2x^2 - y^2 + 3x + 2y = 0$,which subtend a right angle at the origin,always pass through a fixed point $(\alpha, \beta)$,then $(\alpha, \beta) =$

Let $P(\alpha, \beta)$ be a variable point which moves in the $x-y$ plane such that $\frac{PA}{PB} = 2$,where $A(1, 0)$ and $B(0, -1)$. If $M$ and $m$ denote respectively the maximum and minimum value of $\alpha + \beta$,then the value of $[\frac{M}{m}]$ is- (where $[.]$ denotes the greatest integer function)

If a circle $S$ passing through the point $(3,4)$ cuts the circle $x^2+y^2=36$ orthogonally,then the locus of the centre of $S$ is

The hypotenuse of a right-angled triangle has its endpoints at the points $(1, 3)$ and $(-4, 1)$. Find the equations of the legs (perpendicular sides) of the triangle.

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 locus of the image of the point $(2, 3)$ in the line $(2x - 3y + 4) + k(x - 2y + 3) = 0, k \in R$ is a: