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The second degree equation 2x2 + 2y2 - 2x - 6y + 5 = 0 represents
1. a circle
2. an ellipse
3. a point
4. none of the above

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Correct Answer - Option 3 : a point

Concept:

We know that eqn of circle is –

x2 + y2 + 2gx + 2fy + c = 0 …….(1)

whose, centre = (-g, -f) and radius = \(\sqrt {{g^2} + {f^2} - c} \)

Calculation:

Given eqn\({x^2} + {y^2} - x - 3y + \frac{5}{2} = 0\)

On comparing with eqn (1) we get –

2gx = -x, 2fy = -3y,

\(c = \frac{5}{2},g = - \frac{1}{2},\;f = \; - \frac{3}{2}\)

Then, Centre = \(\left( {\frac{1}{2},\frac{3}{2}} \right)\)

Radius = \(\sqrt {{{\left( { - \frac{1}{2}} \right)}^2} + {{\left( { - \frac{3}{2}} \right)}^2} - \left( {\frac{5}{2}} \right)} \)

\( = \sqrt {\frac{1}{4} + \frac{9}{4} - \frac{5}{2}} \)

Radius = \(\sqrt {\frac{{1 + 9 - 10}}{4}} = 0\)

Thus, we can say that a circle whose radius is zero represents a point.

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