Hyperbola 1 Question 6

7. A hyperbola has its centre at the origin, passes through the point $(4,2)$ and has transverse axis of length 4 along the $X$-axis. Then the eccentricity of the hyperbola is

(2019 Main, 9 Jan II)

(a) 2

(b) $\frac{2}{\sqrt{3}}$

(c) $\frac{3}{2}$

(d) $\sqrt{3}$

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Answer:

Correct Answer: 7. (b)

Solution:

  1. Equation of hyperbola is given by

$$ \frac{x^{2}}{a^{2}}-\frac{y^{2}}{b^{2}}=1 $$

$\because$ Length of transverse axis $=2 a=4$

$$ \therefore \quad a=2 $$

Thus, $\frac{x^{2}}{4}-\frac{y^{2}}{b^{2}}=1$ is the equation of hyperbola

$\because$ It passes through $(4,2)$.

$\therefore \frac{16}{4}-\frac{4}{b^{2}}=1 \Rightarrow 4-\frac{4}{b^{2}}=1 \Rightarrow b^{2}=\frac{4}{3} \Rightarrow b=\frac{2}{\sqrt{3}}$

Now, eccentricity,

$$ e=\sqrt{1+\frac{b^{2}}{a^{2}}}=\sqrt{1+\frac{\frac{4}{3}}{4}}=\sqrt{1+\frac{1}{3}}=\frac{2}{\sqrt{3}} $$



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