If tangents `O Q` and `O R` are dawn to variable circles having radius `r` and the center lying on the rectangular hyperbola `x y=1` , then the locus of the circumcenter of triangle `O Q R` is `(O` being the origin). `x y=4` (b) `x y=1/4` `x y=1` (d) none of these

If tangents PA and PB are drawn to circle (x+3)^(2)+(y-2)^(2)=1 from a variable point P on y^(2)=4x, then the equation of the tangent of slope 1 to the locus of the circumcentre of triangle PAB IS

The locus of the center of the circle touching the line 2x-y=1 at (1,1) is (a)x+3y=2 (b) x+2y=2( c) x+y=2 (d) none of these

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.

o is the origin. A and B are variable points on y =x and x +y=0 such that the area of triangle OAB is k^2. The locus of the circumcentre of triangle OAB is

The line L_1-=4x+3y-12=0 intersects the x-and y-axies at Aa n dB , respectively. A variable line perpendicular to L_1 intersects the x- and the y-axis at P and Q , respectively. Then the locus of the circumcenter of triangle A B Q is (a)3x-4y+2=0 (b)4x+3y+7=0 (c)6x-8y+7=0 (d) none of these

ABC is a variable triangle such that A is (1,2) and B and C lie on line y=x+lambda (where lambda is a variable).Then the locus of the orthocentre of triangle ABC is (a)(x-1)^(2)+y^(2)=4(b)x+y=3( c) 2x-y=0 (d) none of these

The tangent at the point P on the rectangular hyperbola xy=k^(2) with C intersects the coordinate axes at Q and R. Locus of the coordinate axes at triangle CQR is x^(2)+y^(2)=2k^(2)(b)x^(2)+y^(2)=k^(2)xy=k^(2)(d) None of these

A variable line y=mx-1 cuts the lines x=2y and y=-2x at points A and B Prove that the locus of the centroid of triangle OAB(O being the origin) is a hyperbola passing through the origin.

If P(x_1, y_1), Q(x_2, y_2), R(x_3, y_3) and S(x_4, y_4) are 4 concyclic points on the rectangular hyperbola xy=c^(2) the coordinates of the orthocentre of the trianglePQR are

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) .