Analytic Geometry – Problem 34: Find the length

Let $H$ be the hyperbola $x^2-\dfrac{y^2}{24}=1$ with right focus $F(5,0)$. The vertical line $x=5$ passes through $F$ and intersects $H$ at two points $A,B$.

Find the length $|AB|$.

(1) Substitute $x=5$ into $x^2-\dfrac{y^2}{24}=1$: $$25-\frac{y^2}{24}=1\Rightarrow \frac{y^2}{24}=24\Rightarrow y^2=576.$$ So the intersection points are $A(5,24)$ and $B(5,-24)$.

(2) Therefore $$|AB|=|24-(-24)|=48.$$

The chord length is $|AB|=48$.