The equation 2x² - 3y² - 6 = 0 represents

Correct Answer is : a hyperbola

Correct Answer is : a hyperbola

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UPSC NDA I 2019 Mathematics Solved Paper

Question : 54 of 120

Marks: +1, -0

The equation

2x^{2} - 3y^{2} - 6 = 0

represents

Solution:

Concept:

A hyperbola centered at the origin has the standard form

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

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

Explanation:

Start with the given equation:

2x^2 - 3y^2 - 6 = 0

Add

6

to both sides:

2x^2 - 3y^2 = 6

Divide every term by

6

\frac{2x^2}{6} - \frac{3y^2}{6} = 1

Simplify the fractions:

\frac{x^2}{3} - \frac{y^2}{2} = 1

This matches

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

with

a^2 = 3

and

b^2 = 2

Therefore, the equation represents a hyperbola.

Answer:

D. a hyperbola.

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