Hyperbola 1 Question 3
4. 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?
(a) $(2 \sqrt{6}, 5)$
(b) $(6,5 \sqrt{2})$
(c) $(4, \sqrt{15})$
(d) $(-6,2 \sqrt{10})$
(2019 Main, 12 Jan I)
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Answer:
Correct Answer: 4. (c)
Solution:
- The vertices of hyperbola are given as $( \pm 2,0)$ and one of its foci is at $(-3,0)$.
$\therefore(a, 0)=(2,0)$ and $(-a e, 0),=(-3,0)$
On comparing $x$-coordinates both sides, we get
$\Rightarrow \quad a=2$ and $-a e=-3$
$\Rightarrow 2 e=3 \Rightarrow e=\frac{3}{2}$
Also, $\quad \frac{9}{4}=1+\frac{b^{2}}{4} \Rightarrow b^{2}=5$
$\because e^{2}=1+\frac{b^{2}}{a^{2}}$
So, equation of the hyperbola is
$$ \frac{x^{2}}{4}-\frac{y^{2}}{5}=1 $$
The point $(6,5 \sqrt{2})$ from the given options does not satisfy the above equation of hyperbola.
