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). (a) `x y=4` (b) `x y=1/4` `x y=1` (d) none of these

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 line from (-1,0) intersects the parabola x^(2)= 4y at A and B. Then the locus of centroid of DeltaOAB is (where O is origin) (a) 3x^(2) -2x =4y (b) 3y^(2) -2y = 4x (c) 3x^(2) +2x=4y (d) none of these

If the intercepts of the variable circle on the x- and y-axis are 2 units and 4 units, respectively, then find the locus of the center of the variable circle.

A variable line through the point P(2,1) meets the axes at a an d b . Find the locus of the centroid of triangle O A B (where O is 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 four concyclic points on the rectangular hyperbola and xy = c^2 , then find coordinates of the orthocentre of the triangle PQR

A circle of radius 'R' passes through the origin O and cuts the axes at A and B,Locus of the centroid of triangle OAB is

The circumcenter of the triangle formed by the line y=x ,y=2x , and y=3x+4 is

The chord P Q of the rectangular hyperbola x y=a^2 meets the axis of x at A ; C is the midpoint of P Q ; and O is the origin. Then A C O is (a) equilateral (b) isosceles (c) right-angled (d) right isosceles

If a circle of radius r is touching the lines x^2-4x y+y^2=0 in the first quadrant at points Aa n dB , then the area of triangle O A B(O being the origin) is (a) 3sqrt(3)(r^2)/4 (b) (sqrt(3)r^2)/4 (c) (3r^2)/4 (d) r^2

If tangents P Qa n dP R are drawn from a variable point P to thehyperbola (x^2)/(a^2)-(y^2)/(b^2)=1,(a > b), so that the fourth vertex S of parallelogram P Q S R lies on the circumcircle of triangle P Q R , then the locus of P is (a) x^2+y^2=b^2 (b) x^2+y^2=a^2 (c) x^2+y^2=a^2-b^2 (d) none of these