If the equation x^2 + y^2 + 2gx + 2fy + 1 = 0 | vedclass

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(A) The general equation of a second-degree curve is given by $ax^2 + 2hxy + by^2 + 2gx + 2fy + c = 0$.Comparing this with the given equation $x^2 + y

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$f^2 + g^2 = 1$

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(A) The general equation of a second-degree curve is given by $ax^2 + 2hxy + by^2 + 2gx + 2fy + c = 0$.Comparing this with the given equation $x^2 + y^2 + 2gx + 2fy + 1 = 0$,we get $a = 1, b = 1, h = 0, c = 1$.The condition for this equation to represent a pair of straight lines is $abc + 2fgh - af^2 - bg^2 - ch^2 = 0$.Substituting the values: $(1)(1)(1) + 2(f)(g)(0) - (1)(f^2) - (1)(g^2) - (1)(0)^2 = 0$.This simplifies to $1 - f^2 - g^2 = 0$.Therefore,$f^2 + g^2 = 1$.

If the equation $x^2 + y^2 + 2gx + 2fy + 1 = 0$ represents a pair of straight lines,then:

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