A circle with centre `(3alpha, 3beta)` and of variable radius cuts the rectangular hyperbola `x^(2)-y^(2)=9a^(2)` at the points `P, Q, S, R`. Prove that the locus of the triangle PQR is `(x-2alpha)^(2)-(y-2beta)^(2)=a^(2)`.

A circle with centre (3alpha, 3beta) and of variable radius cuts the rectangular hyperbola x^(2)-y^(2)=9a^(2) at the points P, Q, S, R . Prove that the locus of the centroid of triangle PQR is (x-2alpha)^(2)-(y-2beta)^(2)=a^(2) .

A circle of variable radius cuts the rectangular hyperbola x^(2)-y^(2)=9a^(2) in points P, Q, R and S. Determine the equa- tion of the locus of the centroid of triangle PQR.

A circle with centre (3a,3b) and with variable radius cuts the hyperbola x^(2)-y^(2)=36 at the points A,B,C and D . If the locus of the centroid of Delta ABC is (x-k_(1)a)^(2)-(y-k_(2)b)^(2)=k_(3) . then value of k_1, k_2 and k_3

If (alpha,beta) is the centre of the circle which touches the curve x^(2)+xy-y^(2)=4 at (2,2) and the line 3x-y+6=0 ,then the value of alpha+beta is equal to

The tangent at the point (alpha, beta) to the circle x^2 + y^2 = r^2 cuts the axes of coordinates in A and B . Prove that the area of the triangle OAB is a/2 r^4/|alphabeta|, O being the origin.

If two perpendicular tangents can be drawn from the point (alpha,beta) to the hyperbola x^(2)-y^(2)=a^(2) then (alpha,beta) lies on

If the tangent to the curve 2y^(3)=ax^(2)+x^(3) at the point (a,a) cuts off intercept alpha and beta on the co-ordinate axes , (where alpha^(2)+beta^(2)=61 ) then a^(2) equals ______

IF (alpha, beta) is a point on the chord PQ of the circle x^(2)+y^(2)=25 , where the coordinates of P and Q are (3, -4) and (4, 3) respectively, then

Prove that the locus of the middle-points of the chords of the hyperbola (x^(2))/(a^(2))-(y^(2))/(b^(2))=1 which pass through a fixed point (alpha, beta) is a hyperbola whose centre is ((alpha)/(2), (beta)/(2)) .

If x=r sin alpha cos beta, y= r sin alpha sin beta and z= r cos alpha , prove that x^(2)+y^(2) + z^(2) = r^(2).