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 staright 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

We have `(ax^(2)+by^(2)+c)(x^(2)-5xy+6y^(2))=0`
`therefore" "ax^(2)+by^(2)+c=0 or x^(2)-5xy+6y^(2)=0`
From `x^(2)-5xy+6y^(2)=0` we have `(x-2y)(x-3y)=0`, which represents a pair of straight lines.
If c = 0 and a and b have same sign then we have `ax^(2)+by^(2)=0,`
which is possible. Only if `(x,y)-=(0,0).`
If a = b and c has sign opposite to a, we have `ax^(2)+ay^(2)=-c or x^(2)+y^(2)=-c//a`, which represents the circle.
If a and b have same sign opposite to that of c, then `ax^(2)+by^(2)=-c` is equation of ellipse.