If $P = (1, 0) ; Q = (-1, 0) \,\,ane\,\, R = (2, 0)$ are three given points, then the locus of the points $S$ satisfying the relation, $SQ^2 + SR^2 = 2 SP^2$ is :

The number of integral points (integral point means both the coordinates should be integer) exactly in the interior of the triangle with vertices $(0, 0), (0, 21)$ and $(21, 0)$, is

The equation of the line which makes right angled triangle with axes whose area is $6$ sq. units and whose hypotenuse is of $5$ units, is

If the straight line $ax + by + c = 0$ always passes through $(1, -2),$ then $a, b, c$ are

Let the circumcentre of a triangle with vertices $A ( a , 3), B ( b , 5)$ and $C ( a , b ), ab >0$ be $P (1,1)$. If the line $AP$ intersects the line $BC$ at the point $Q \left( k _{1}, k _{2}\right)$, then $k _{1}+ k _{2}$ is equal to.

Let $PQR$ be a right angled isosceles triangle, right angled at $P\, (2, 1)$. If the equation of the line $QR$ is $2x + y = 3$, then the equation representing the pair of lines $PQ$ and $PR$ is