Question

The curve represented by a second degree curve x² - 2y² + 2x -8y +1 = 0 is
1. circle
2. ellipse
3. parabola
4. hyperbola

Answer 1

Correct Answer - Option 4 : hyperbola

Concept:

Identification of curves represented by the general equation of the second degree-

Let a general equation of second degree in x and y be ax2 + 2hxy + by2 + 2gx + 2fy + c = 0       ----(A)

Then this equation represents,

1. a parabola if Δ ≠ 0 and h2 = ab
2. an ellipse if  Δ ≠ 0 and h2 < ab
3. a hyperbola if Δ ≠ 0 and h2 > ab
4. a pair of straight line or empty set if  Δ = 0 and h2 ≥ ab
5. a unique point if  Δ = 0 and h2 < ab
6. a circle if a = b ≠ 0, h = 0 and g2 + f2 -ac > 0

Calculation:

The given equation is x2 - 2y2 + 2x -8y +1 = 0

comparing it with the equation (A) we get, a = 1, h = 0, b = -2, g = 1, f = - 4, c = 1

Here, Δ = abc + 2fgh - af2 - bg2 -ch2

Δ = 1 × -2 × 1 + 2 × 0 - 1 × (-4)2 - (-2)× (1)2 - 1 × 0

Δ = -2 -16 + 2

Δ = -16 ≠ 0

and h² = 0, ab = 1 × (-2) = -2 ⇒ h² > ab

Hence, the given equation represents a hyperbola.