Question

If the co-ordinates of the vertices of a hyperbola are $(9,2)$ and $(1,2)$ and the distance between its two foci is $10$, then the point $(15,12)$ lies ________ of the hyperbola.

• A. Inside
• B. Outside
• C. On
• D. Cannot be determined from the given information

Answer 1

The correct option is B Outside

Let the hyperbola be $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$

Distance between vertices of hyperobla is $\sqrt{(9-1{)}^2+(2-2{)}^2}=8$

Thus, $2a=8$

$\Rightarrow a=4$

Distance between foci $=$ $2c=10$,

where $b^2=c^2-a^2$ ......(i)

Thus $c=5$

Putting this value in equation (i),

$b^2=25-16$

$\Rightarrow b=3$

The hyperbola is $\frac{x^2}{16}-\frac{y^2}{9}=1$

By putting the point if it less than $0$, then the point is inside the hyperbola.

If it is greater than $0$, then it is outside the hyperbola and if it is equal to $0$, then it is on the hyperbola.

So, $\frac{225}{16}-\frac{144}{9}-1$ which is less than $0$, hence the point lies on the hyperbola.