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 PdotO Q=p^2` , then find the locus of the point `Qdot`

If P be any point on the plane ln+my+nz=P and Q be a point on the line OP such that (OP)(OQ)=P^(2). The locus of point Q is

If P be a point on the lane lx+my+nz=p and Q be a point on the OP such that OP. OQ=p^2 show that the locus of the point Q is p(lx+my+nz)=x^2+y^2+z^2 .

The tangent at any point P on the circle x^2+y^2=4 meets the coordinate axes at Aa n dB . Then find the locus of the midpoint of A Bdot

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

From any point on the line y=x+4 tangents are drawn to the auxiliary circle of the ellipse x^(2)+4y^(2)=4. If P and Q are the points of contact and A and B are the corresponding points of P and Q on he ellipse,respectively,then find the locus of the midpoint of AB.

The tangent at any point P on the circle x^2 + y^2 = 2 cuts the axes in L and M . Find the locus of the middle point of LM .

Let P be a point on the hyperbola x^(2)-y^(2)=a^(2), where a is a parameter,such that P is nearest to the line y=2x. Find the locus of P.

From any point P on the line x = 2y perpendicular is drawn on y = x. Let foot of perpendicular is Q. Find the locus of mid point of PQ.

If P is point on the parabola y ^(2) =8x and A is the point (1,0), then the locus of the mid point of the line segment AP is

If A (3,1) and B (-5,7) are any two given points, If P is a point one the line y = x such that PA + PB is minimum then P is