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

A tangent to the hyperbola y = (x+9)/(x+5) passing through the origin is

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).

A variable line passing through point P(2,1) meets the axes at A and B . Find the locus of the circumcenter of triangle O A B (where O is the origin).

Let a given line L_1 intersect the X and Y axes at P and Q respectively. Let another line L_2 perpendicular to L_1 cut the X and Y-axes at Rand S, respectively. Show that the locus of the point of intersection of the line PS and QR is a circle passing through the origin

A variable line intersects the co-ordinate axes at A and B and passes through a fixed point (a, b) .then the locus of the vertex C of the rectangle OACB where O is the origin is

Find the locus of the middle points of the chords of the circle x^(2) + y^( 2) = 4 ( y + 1) drawn through the origin.

If m is a variable, then prove that the locus of the point of intersection of the lines x/3-y/2=m and x/3+y/2=1/m is a hyperbola.

The equation of the straight line through the intersection of line 2x+y=1 and 3x+2y=5 passing through the origin, is:

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) .

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