SATHEE: Hyperbola 1 Question 12

Hyperbola 1 Question 12

13. For hyperbola $\frac{x^{2}}{\cos^{2} α} - \frac{y^{2}}{\sin^{2} α} = 1$ , which of the following remains constant with change in ’ $α$ ‘? (2003, 1M)

(a) Abscissae of vertices

(b) Abscissae of foci

(d) Directrix

$(2006, 3 M)$

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Answer:

Correct Answer: 13. (b)

Solution:

Given equation of hyperbola is $\frac{x^{2}}{\cos^{2} α} - \frac{y^{2}}{\sin^{2} α} = 1$

Here, $a^{2} = \cos^{2} α$ and $b^{2} = \sin^{2} α$

[i.e. comparing with standard equation $\frac{x^{2}}{a^{2}} - \frac{y^{2}}{b^{2}} = 1$ ]

We know that, foci $= (\pm a e, 0)$

where, $a e = \sqrt{a^{2} + b^{2}} = \sqrt{\cos^{2} α + \sin^{2} α} = 1$ $\Rightarrow$ Foci $= (\pm 1, 0)$

where, vertices are $(\pm \cos α, 0)$ .

Eccentricity, $a e = 1$ or $e = \frac{1}{\cos α}$

Hence, foci remains constant with change in $α$ .

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Hyperbola 1 Question 12

13. For hyperbola x2cos2⁡α−y2sin2⁡α=1, which of the following remains constant with change in ’ α ‘? (2003, 1M)

Answer:

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13. For hyperbola $\frac{x^{2}}{\cos^{2} α} - \frac{y^{2}}{\sin^{2} α} = 1$ , which of the following remains constant with change in ’ $α$ ‘? (2003, 1M)