Question

Let a and b be non-zero and real numbers. Then, the equation $(ax^2 + by^2 + c) \, ( x^2 - 5xy + 6y^2) = 0$ represents

• A. four straight lines, when c = 0 and a, b are of the same sign
• B. two straight lines and a circle, when a = b and c is of sign opposite to that of a
• C. 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
• D. a circle and an ellipse, when a and b are of the same sign and c is of sign opposite to that of a

Answer 1

Answer: (B)

two straight lines and a circle, when a = b and c is of sign opposite to that of a

Answer 2

Let a and b be non-zero real numbers
Therefore the given equation
$(ax^2 + by^2 + c) \, ( x^2 - 5xy + 6y^2) = 0$ implies either
$x^2 - 5 x y + 6 y^2 = 0$
$\Rightarrow (x - 2y) \, (x - 3y) = 0$
$\Rightarrow x = 2y$
and x = 3y
represent two straight lines passing through origin or
$ax^2 + by^2 + c = 0$ when c = 0 and a and b are of same
signs, then
$ax^2 + by^2 + c = 0$
x = 0

and y =- 0
which is a point specified as the origin.
When, a = b and c is of sign opposite to that of a
$ax^2 + by^2 + c = 0$ represents a circle.
Hence, the given equation
\hspace22mm $(ax^2 + by^2 + c) \, (x^2 - 5xy + 6 y^2 ) = 0$
may represent two straight lines and a circle.