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

Options

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

B

Explanation

When a and b are of the same sign and c = 0, given equation reduces to
`(ax^(2)+by^(2))(x^(2)-5xy+6y^(2))=0`
`rArr x^(2)-5xy+6y^(2)=0 [ :' ax^(2)+by^(2)!=0 "when a and b are of the same sign"]`
`rArr (x-2y)(x-3y)=0 rArr x=2y, x=3y`
So, the given equation represents a pair of straight lines. When a = b and the sign of c is opposite to that of a. The given equation reduces to
`(ax^(2)+ay^(2)+c)(x^(2)-5xy+6y^(2))=0`
`rArr ax^(2)+ay^(2)+c=0 or x^(2)-5xy+6y^(2)=0`
`rArr x^(2)+y^(2)=-(c)/(a), x=2y, x=3y`
So, the given equation represents a circle of radius `sqrt((-c)/(a))` and a pair of straight lines.