Finding Foci of a Conic

Given Equation: \( x^2 + 2x - 4y^2 + 8y - 7 = 0 \)

Step 1: Complete the square

⇒ \( (x + 1)^2 - 4(y - 1)^2 = 4 \)

Rewriting: \( \frac{(x + 1)^2}{4} - \frac{(y - 1)^2}{1} = 1 \)

This is a horizontal hyperbola with:

• Center: \( (-1, 1) \)
• \( a^2 = 4 \), \( b^2 = 1 \)
• \( c = \sqrt{a^2 + b^2} = \sqrt{5} \)

✅ Foci: \( (-1 \pm \sqrt{5},\ 1) \)