Tangents are drawn to the hyperbola 4x^2-y^2=36 at the points P and Q.

Question

Tangents are drawn to the hyperbola $4x^2-y^2=36$ at the points $P$ and $Q$. If these tangents intersect at the point $T(0,3)$ then the area (in sq. units) of $	riangle PTQ$ is:

Accepted Answer

$\dfrac{x^2}9-\dfrac{y^2}{36}=1$. Chord of contact from $T(0,3)$: $-\dfrac{3y}{36}=1\Rightarrow y=-12$.

Substituting into the hyperbola: $4x^2-144=36\Rightarrow x^2=45\Rightarrow x=\pm3\sqrt5$.

$$P=(3\sqrt5,-12),\ Q=(-3\sqrt5,-12);\qquad PQ=6\sqrt5,\ \text{height}=|3-(-12)|=15.$$

$$\text{Area}=\frac12(6\sqrt5)(15)=45\sqrt5.$$

Solutions · 1