The normals at P,Q and R are concurrent and PQ meets the diameter through R on the directrix x=-a . Prove that PQ touches [or PQ enveleopes] the parabola `y^2+16a(x+a)=0`.

The normals at P,R,R on the parabola y^(2)=4ax meet in a point on the line y=c . Prove that the sides of the triangle PQR touch the parabola x^(2)=2cy

The normal at P(8,8) to the parabola y^(2)=8x cuts it again at Q then PQ

The normal at P(2,4)to y^2=8x meets the parabola at Q. Then the radius of the circle on normal chord PQ as diameter is.

Consider the parabola with vertex (1/2, 3/4) and the directrix y = 1/2 . Let P be the point where the parabola meets the line x = - 1/2 . If the normal to the parabola at P intersects the parabola again at the point Q, then (PQ)^2 is equal to :

Consider the parabola with vertex (1/2, 3/4) and the directrix y = 1/2 . Let P be the point where the parabola meets the line x = - 1/2 . If the normal to the parabola at P intersects the parabola again at the point Q, then (PQ)^2 is equal to :

If the normal to the parabola y^(2)=12x at the point P(3,6) meets the parabola again at the point Q,then equation of the circle having PQ as a diameter is

PQ is a double ordinate of the ellipse x^(2)+9y^(2)=9, the normal at P meets the diameter through Q at R .then the locus of the mid point of PR is

The normal to the parabola y^(2)=4x at P(9, 6) meets the parabola again at Q. If the tangent at Q meets the directrix at R, then the slope of another tangent drawn from point R to this parabola is

Let PQ and RS be tangent at the extremities of the diameter PR of a circle of radius r. If P S and RQ intersect at a point X on the circumference of the circle,then prove that 2r=sqrt(PQ xx RS).