If the vertices of a hyperbola be at
Question:

If the vertices of a hyperbola be at $(-2,0)$ and $(2,0)$ and one of its foci be at $(-3,0)$, then which one of the following points does not lie on this hyperbola?

1. (1) $(-6,2 \sqrt{10})$

2. (2) $(2 \sqrt{6}, 5)$

3. (3) $(4, \sqrt{15})$

4. (4) $(6,5 \sqrt{2})$

Correct Option: , 4

Solution:

Let the points are,

$A(2,0), A^{\prime}(-2,0)$ and $S(-3,0)$

$\Rightarrow$ Centre of hyperbola is $O(0,0)$

$A A^{\prime}=2 a \Rightarrow 4=2 a \Rightarrow a=2$

$\because \quad$ Distance between the centre and foci is $a e$.

$\therefore \quad O S=a e \Rightarrow 3=2 e \Rightarrow e=\frac{3}{2}$

$\Rightarrow \quad b^{2}=a^{2}\left(e^{2}-1\right)=a^{2} e^{2}-a^{2}=9-4=5$

$\Rightarrow \quad$ Equation of hyperbola is $\frac{x^{2}}{4}-\frac{y^{2}}{5}=1$….(1)

$\because \quad(6,552)$ does not satisfy eq (1).

$\therefore \quad(6,5 \sqrt{2})$ does not lie on this hyperbola.