Two fixed points `A` and `B` are taken on the coordinates axes such that `O A=a` and `O B=b` . Two variable points `A '` and `B '` are taken on the same axes such that `O A^(prime)+O B^(prime)=O A+O Bdot` Find the locus of the point of intersection of `A B^(prime)` and `A^(prime)Bdot`

B is a variable point on y^(2)=4ax. The line OB(O: origin is extended to the point M such that OM=lambda0B. The locus of M is

Let L_1=0a n dL_2=0 be two fixed lines. A variable line is drawn through the origin to cut the two lines at R and SdotPdot is a point on the line A B such that ((m+n))/(O P)=m/(O R)+n/(O S)dot Show that the locus of P is a straight line passing through the point of intersection of the given lines R , S , R are on the same side of O)dot

Using determinants prove that the points (a ,\ b),\ (a^(prime),\ b^(prime))a n d\ (a-a^(prime),\ b-b^(prime)) are collinear if a b^(prime)=a^(prime)bdot

The locus of the centre of the circle passing through the origin O and the points of intersection A and B of any line through (a, b) and the coordinate axes is

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

If a circle of radius R passes through the origin O and intersects the coordinates axes at A and B, then the locus of the foot of perpendicular from O on AB is:

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