Let $t_{1} a n d t_{2}$ be the parameters of 2 points on a parabola. What is the value of $t_{1} t_{2}$ if tangents at these points are at right angle to each other?
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Solution

We are given the parameters of the 2 points A and B.

Also given taht the tangents at these points make $90^{\circ}$ with each other.

This implies that the product of slopes is -1.

slopes of tangent at A= $(a t_{1}^{2}, 2 a t_{1}^{□}) = \frac{2 a t_{1}}{a t_{1}^{2}} = \frac{2}{t_{1}}$

Similarly slope at B= $\frac{2}{t_{2}}$

$∴ \frac{2}{t_{1}} . \frac{2}{t_{2}} = - 1$

i.e., $t_{1} . t_{2} = - 1$

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