Let a and b be bonzero real numbers. Then the equation `(ax^(2)+by^(2)+c)(x^(2)-5xy+6y^(2))=0` 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 circle, when `a=b` 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`

Explanation

Let `a` and `b` 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`
`implies(x-2y)(x-3y)=0`
`implies x=2y`
and `x=3y`
represent two straight line 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,
`(ax^(2)+by^(2)+c)(x^(2)-5xy+6y^(2))=0`.
may represent two straight lines and a circle.