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.
There is a geometry argument without any algebra for this problem.
Denote the two extremities of the focal chord as A and B, let M be the midpoint of the segment AB. Draw a segment MH which is parallel to the x-axis, crossing the directrix at H.
Draw AD and BE parallel to the x-axis, crossing the directrix at D, E respectively. Connect AH and BH. Let F be the focal point.
Then we have AB=AF+FB=AD+BE=2MH. So we have AM=MB=MH. So AHB is a right triangle. We need only to show that AH and BH are tangent lines. This is quite easy using the optical properties of the parabola.
For two negtive number $x_1,x_2$, we can get $\sqrt{x_1x_2}$ of course.
If some one believes that $\sqrt{x_1x_2}$ is the geometric mean of two negtive number $x_1,x_$, then we will get the conclusion that in some cases the "mean" is greater than every memeber. Then how can we call it the "mean"?
If some one believes that $-\sqrt{x_1x_2}$ is the geometric mean of two negtive number $x_1,x_$, then how can we explain the inequality AM >= GM?