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Let 'a' and 'b' be non-zero real numbers...

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

Text Solution

Verified by Experts

The correct Answer is:
B
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^2-5xy+6y^2=0`
`rArr (x-2y(x-3y)=0`
`rArr x=2y and x=3y` represents two straight lines passing through origin. or `ax^2+by^2+c=0`.
Which `c=0` and a and b are of same signs , then
`ax^2+by^2+c=0`
`rArr x=0 and y=0` which is a point specified as the origin.
When `a=b` and c is of sign opposite to that a , then `ax^2+by^2+c=0` represents a circle.
Hence , the given equation.
`(ax^2+by^2+c)(x^2-5xy+6y^2)=0`
may represents two straight lines and a circle.
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