JEE Advance - Mathematics (2008 - Paper 1 Offline - No. 20)
Let a and b be non-zero real numbers. Then, the equation
$$(a{x^2} + b{y^2} + c)({x^2} - 5xy + 6{y^2}) = 0$$ represents :
$$(a{x^2} + b{y^2} + c)({x^2} - 5xy + 6{y^2}) = 0$$ represents :
four straight lines, when c = 0 and a, b are of the same sign
two straight lines and a circle, when a = b, and c is of sign opposite to that of a
two straight lines and a hyperbola, when a and b are of the same sign and c is of sign opposite to that of a
a circle and an ellipse, when a and b are of the same sign and c is of sign opposite to that of a
Explanation
Let a and b be non-zero real numbers.
Therefore, the given equation $$(a{x^2} + b{y^2} + c)({x^2} - 5xy + 6{y^2}) = 0$$ implies either
$${x^2} - 5xy + 6{y^2} = 0$$
$$(x - 2y)(x - 3y) = 0$$
$$x = 2y$$ and $$x = 3y$$ represent two straight line passing through origin or $$a{x^2} + b{y^2} + c = 0$$ when c = 0 and a and b are of same signs then
$$a{x^2} + b{y^2} + c = 0$$
$$y=0$$
Which is a point specified as the origin. When a = b and c is of sign opposite to that of a $$a{x^2} + b{y^2} + c = 0$$ represent a circle.
Hence, the given equation,
$$(a{x^2} + b{y^2} + c)({x^2} - 5xy + 6{y^2}) = 0$$
May represent two straight lines and a circle.
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