. What is the distance between the two foci of the hyperbola 25x² - 75y ²= 225 ?

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Question

This question was previously asked in

NDA-I (Mathematics) Official Paper (Held On: 13 Apr, 2025)

Answer 1

Calculation:

Given,

Hyperbola equation: \(25x^{2} - 75y^{2} = 225\)

Divide both sides by 225 to obtain standard form:

\(\frac{25x^{2}}{225} - \frac{75y^{2}}{225} = 1 \;\Longrightarrow\; \frac{x^{2}}{9} - \frac{y^{2}}{3} = 1\)

Thus, \(a^{2} = 9\) and \(b^{2} = 3\).

Compute \(c\) from \(c^{2} = a^{2} + b^{2}\):

\(c^{2} = 9 + 3 = 12 \;\Longrightarrow\; c = \sqrt{12} = 2\sqrt{3}.\)

The foci are at \((\pm c,\,0)\), so the distance between them is \(2c\):

\(2c = 2 \times 2\sqrt{3} = 4\sqrt{3}.\)

∴ The distance between the two foci is \(4\sqrt{3}\) units.

Hence, the correct answer is Option 2.

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