Let a and b be non-zero real numbers. Then, the equation ( a x 2 + b y 2 + c ) ( x 2 - 5 x y + 6 y 2 ) = 0 represents [2008]

Question

Let $a$ and $b$ be non-zero real numbers. Then, the equation $(a x^{2} + b y^{2} + c) (x^{2} - 5 x y + 6 y^{2}) = 0$ represents [2008]

Smart Achievers · Accepted Answer

(2)

$∵$ $x^{2} - 5 x y + 6 y^{2} = 0$

$\Rightarrow (x - 3 y) (x - 2 y) = 0$

$∴$ $x^{2} - 5 x y + 6 y^{2} = 0 represents a pair of straight lines given by$ $x - 3 y = 0 and x - 2 y = 0 .$

$Also a x^{2} + b y^{2} + c = 0 will represent a circle if a = b$ $and c is of sign opposite to that of a .$

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