Question

Which of the following is equation of hyperbola with vertices \((\pm 2,0)\) and foci \((\pm 3,0)\)?

• A. \(\dfrac{x^2}{4}-\dfrac{y^2}{9}=1\)
• B. \(\dfrac{x^2}{4}+\dfrac{y^2}{9}=1\)
• C. \(\dfrac{x^2}{4}-\dfrac{y^2}{5}=1\)
• D. \(\dfrac{x^2}{5}-\dfrac{y^2}{4}=1\)

Hint:

For a hyperbola \(\frac{x^2}{a^2}-\frac{y^2}{b^2}=1\), vertices give \(a\), foci give \(c\), and use \(c^2=a^2+b^2\).

Answer 1

The Correct Option is C

Concept:
For a hyperbola with transverse axis along the \(x\)-axis, the standard equation is: \[ \frac{x^2}{a^2}-\frac{y^2}{b^2}=1 \] The vertices are: \[ (\pm a,0) \] and the foci are: \[ (\pm c,0) \] where: \[ c^2=a^2+b^2 \]

Step 1: Use the vertices.
Given vertices are: \[ (\pm 2,0) \] Therefore, \[ a=2 \] \[ a^2=4 \]

Step 2: Use the foci.
Given foci are: \[ (\pm 3,0) \] Therefore, \[ c=3 \] \[ c^2=9 \]

Step 3: Find \(b^2\). \[ c^2=a^2+b^2 \] \[ 9=4+b^2 \] \[ b^2=5 \]

Step 4: Write the equation. \[ \frac{x^2}{a^2}-\frac{y^2}{b^2}=1 \] \[ \frac{x^2}{4}-\frac{y^2}{5}=1 \] \[ \therefore \text{Correct Answer is (C)} \]