Let `L_(1) =0 " and " L_(2)=0` be two fixed lines. A variable line is drawn through the origin to cut the two lines at R and S. P is a point on the line AB such that (m+n)/OP=m/OR+n/OS. Show that the locus of P is a straight line passing through the point of intersection of the given lines (R,S,P are on the same side of O).

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

Find the equation of the straight line passing through the point of intersection of the straight lines x+2y+3=0 and 3x+4y+7=0 and parallel to the straight line y=-(5)/(8)x .

Given n straight line and a fixed point O, a straight line is drawn through O meeting these lines in the points R_1, R_2, R_3……R_n . If R is the point on the line such that n/(OR)=1/(OR_1)+1/(OR_2)+....+1/(OR_n) , then show that the locus of R is a straight line.

If P is any point on the plane l x+m y+n z=pa n dQ is a point on the line O P such that O P.O Q=p^2 , then find the locus of the point Qdot

If P be any point on the plane lx+my+nz=p and Q be a point on the line OP such that OP.OQ=p^(2) , then find the locus of the point Q.

A variable line cuts n given concurrent straight lines at A_1,A_2...A_n such that sum_(i=1)^n 1/(OA_i) is a constant. Show that it always passes through a fixed point, O being the point of intersection of the lines

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).

Two straight lines pass through the fixed points (+-a, 0) and have slopes whose products is pgt0 Show that the locus of the points of intersection of the lines is a hyperbola.

Find the locus of the circumcenter of a triangle whose two sides are along the coordinate axes and the third side passes through the point of intersection of the line a x+b y+c=0 and l x+m y+n=0.

A variable straight line passes through the point of intersection of the staright lines x/a+y/b=1 and x/b+y/a=1 and intersects the axes at P and Q.Find the locus of mid-point of Pq.