RHS show that the tangents at the extremities of any focal chord of a parabola intersect at right angles at directrix.

My work:

Consider ends of focal chord as P( )and Q( ) .

now point of intersection of tangents are( ).now we know hence point of intersection are .clearly this lies on directrix.

but my second method creates a problem .we know that x-co-ordinate of point of intersection of tangents at P and Q on the parabola is the GEOMETRIC MEAN of the x coordinate of P and Q and y coordinate is the ARITHMETIC MEAN of y coordinates(by definition of book) .therefore for point of intersection I take the GM of x coordinate as hence answer must be (x coordinate)as .

Where is the error.is the definition of book is true for GM and AM for some special case.