In Question 4(b), I am struggling to understand the reason on how the answer key got its values and why they make sense. I do understand the concepts of hyperbolas and asymptotes, however, I do not understand how $\sqrt{10}$ or dividing plays a part. Thanks!
(b) $P$ is a point in the hyperbola $\frac{x^2}{4} -\frac{y^2}{36}=1$ in the right branch of the hyperbola. From $P$, lines are drawn perpendicular to the asymptotes. The points where the perpendiculars meet the asymptotes are $A$ and $B$, as shown on the diagram. Show that the product of the lengths of the two segments $PA$ and $PB$ is constant, and state its value.
Question Booklet: http://www.nzqa.govt.nz/assets/scholarship/2016/93202-qbk-2016.pdf
Answer Key: http://www.nzqa.govt.nz/assets/scholarship/2016/93202-ass-2016.pdf
Top Scholar's Paper :http://www.nzqa.govt.nz/assets/scholarship/2016/93202-exp-2016-topscholar.pdf
They used the formula for the distance from a point to a line (which is also the perpendicular distance) to find the length of AP and AB which is where the $\sqrt{10}$ comes from. https://en.wikipedia.org/wiki/Distance_from_a_point_to_a_line