Question

If the parabola $y=(a-b)x^2+(b-c)x+(c-a)$ touches the x - axis then the line ax + by + c = 0

• A. Always passes through a fixed point
• B. represents the family of parallel lines
• C. always perpendicular to x-axis
• D. always has negative slope

Answer 1

The correct option is A Always passes through a fixed point

Solving equation of parabola with x-axis ( y=0)

We get $(a-b)x^2+(b-c)x+(c-a)=0,$ which should have two equal values of x, as x-axis touches the parabola $\Rightarrow (b-c{)}^2-4(a-b\left)\right(c-a)=0$
$\Rightarrow (b+c-2a{)}^2=0\Rightarrow -2a+b+c=0\Rightarrow ax+by+c=0$ always passes through (-2, 1) (If we substitute (-2,1) in the equation of the line, then we get -2a+b+c=0, which is true)