Question

Find the focus of the parabola y^2=1/2x^2-x+1/2?

Answer 1

(1) : The given eqn. represents a pair of lines passing through the point `(1,0).`
(2) : The individual lines of the pair are,
`x-sqrt2y-1=0, and, x+sqrt2y-1=0.`
(3) : The given eqn. does not represent a parabola, hence, no focus.

Explanation:

Assuming that, the given eqn. is, `y^2=1/2x^2-x+1/2.`

Multiplying by `2," we get, "2y^2=x^2-2x+1=(x-1)^2. i.e.,`

`(x-1)^2=2y^2 rArr x-1=pmsqrt2*y.`

This means that, the given eqn. represents a pair of lines.

The Individual Lines of this pair are,

`l_1 : x-sqrt2y-1=0, and, l_2 : x+sqrt2y-1=0.`

Their point of intersection is `(1,0).`

Thus, the given eqn. represents a pair of lines passing

through the point `(1,0).`