Originally Posted by

**ayushdadhwal** RHS show that the tangents at the extremities of any focal chord of a parabola $\displaystyle y^2=4ax$ intersect at right angles at directrix.

My work:

Consider ends of focal chord as P($\displaystyle at_1^2,2at_1$)and Q($\displaystyle at_2^2,2at_2$) .

now point of intersection of tangents are($\displaystyle at_1t_2 ,2a(t_1+t_2$).now we know $\displaystyle t_1t_2=-1$ hence point of intersection are $\displaystyle (-a,a(t_1+t_2))$.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 $\displaystyle \sqrt( a^2t_1^2t_2^2)$ hence answer must be $\displaystyle a$ (x coordinate)as $\displaystyle a>0$.

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