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The second degree equation 2x2 + 2y2 - 5x -7y -3 = 0 represents
1. circle
2. parabola
3. ellipse
4. hyperbola

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Correct Answer - Option 1 : circle

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 + f-ac > 0


Calculation:

The given equation is 2x2 + 2y2 - 5x - 7y - 3 = 0

comparing it wth the equation (A) we get, a = 2, h = 0, b = 2, g = -(5/2), f = - (7/2), c = -3

Here, a = b = 2 (>0), h = 0 and

Δ = abc + 2fgh - af2 - bg2 -ch2

Δ = 2 × 2 × (-3) + 2 × 0 - 2 × (-5/2)2 - 2 × (-7/2)2 - (3) × 0

Δ = -12 - (25/2) - (49/2)

Δ = -39 < 0

Hence, the given equation represents a circle.

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