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 ax^{2} + 2hxy + by^{2} + 2gx + 2fy + c = 0 ----(A)

Then this equation represents,

- a parabola if Δ ≠ 0 and h
^{2} = ab
- an ellipse if Δ ≠ 0 and h2 < ab
- a hyperbola if Δ ≠ 0 and h2 > ab
- a pair of straight line or empty set if Δ = 0 and h2 ≥ ab
- a unique point if Δ = 0 and h
^{2} < ab
- a circle if a = b ≠ 0, h = 0 and g
^{2} + f^{2 }-ac > 0

**Calculation:**

The given equation is 2x^{2} + 2y^{2} - 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 - af**^{2} - bg^{2} -ch^{2}

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 r**epresents a circle.**