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Question

Let a and b be non-zero and real numbers. Then, the equation
(ax2+by2+c)(x2−5xy+6y2)=0 represents

A
Four straight lines, when c=0 and a, b are of the same sign
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B
Two straight lines and a circle, when a=b and c is of sign opposite to that of a
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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
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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
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Solution

The correct option is B Two straight lines and a circle, when a=b and c is of sign opposite to that of a
Let a and b non-zero real numbers.
Therefore the given equation
(ax2+by2+c)(x25xy+6y2)=0 implies either
x25xy+6y2=0
(x2y)(x3y)=0
x=2y
And x=3y
Represent two straight lines passing through origin or ax2+by2+c=0 when c=0 and a and b are of same signs, then
ax2+by2+c=0x=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, ax2+by2+c=0 represents a circle.
Hence, the given equation,
(ax2+by2+c)(x25xy+6y2)=0
May represent two straight lines and a circle.


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