Problem 14P

14. Tangent lines $T_{1}$ and $T_{2}$ are drawn at two points $P_{1}$ and $P_{2}$ on the parabola $y=x^{2}$ and they intersect at a point $P$. Another tangent line $T$ is drawn at a point between $P_{1}$ and $P_{2}$; it intersects $T_{1}$ at $Q_{1}$ and $T_{2}$ at $Q_{2}$. Show that

$$ \frac{\left|P Q_{1}\right|}{\left|P P_{1}\right|}+\frac{\left|P Q_{2}\right|}{\left|P P_{2}\right|}=1 $$

Step-by-Step Solution