A rectangular hyperbola is drawn by rotating the axis of hyperbola x2 - y2 - a2 in clockwise direction by 45- The length of the latus rectum of the rectangular hyperbola is

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Question

A rectangular hyperbola is drawn by rotating the axis of hyperbola $x^{2} - y^{2} = a^{2}$ in clockwise direction by $45^{\circ} .$ The length of the latus rectum of the rectangular hyperbola is

2 a

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4 a

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a

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\sqrt{2} a

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Solution

The correct option is A $2 a$
Given, Hyperbola ; $x^{2} - y^{2} = a^{2}$
The equation of rectangular hyperbola rotating this hyperbola
in clockwise direction by $45^{\circ} .$
$\Rightarrow x y = c^{2}$
where $c^{2} = \frac{a^{2}}{2} \Rightarrow c = \frac{a}{\sqrt{2}}$
The length of latus rectum to $x y = c^{2}$ is $2 \sqrt{2} c$
$\Rightarrow 2 \sqrt{2} c = 2 \sqrt{2} \times \frac{a}{\sqrt{2}} = 2 a$
Hence, the correct answer is option (a).

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A rectangular hyperbola is drawn by rotating the axis of hyperbola x2−y2=a2 in clockwise direction by 45∘. The length of the latus rectum of the rectangular hyperbola is

The correct option is A 2a Given, Hyperbola ; x2−y2=a2 The equation of rectangular hyperbola rotating this hyperbola in clockwise direction by 45∘. ⇒xy=c2 where c2=a22⇒c=a√2 The length of latus rectum to xy=c2 is 2√2c ⇒2√2c=2√2×a√2=2a Hence, the correct answer is option (a).

A rectangular hyperbola is drawn by rotating the axis of hyperbola $x^{2} - y^{2} = a^{2}$ in clockwise direction by $45^{\circ} .$ The length of the latus rectum of the rectangular hyperbola is