SATHEE: Hyperbola 1 Question 6

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

(d) $\sqrt{3}$

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Answer:

Correct Answer: 7. (b)

Solution:

Equation of hyperbola is given by

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

$∵$ Length of transverse axis $= 2 a = 4$

$∴ a = 2$

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

$∵$ It passes through $(4, 2)$ .

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

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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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

Answer:

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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