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Question

If the parabola y=(a−b)x2+(b−c)x+(c−a) touches the x - axis then the line ax + by + c = 0

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

The correct option is A Always passes through a fixed point
Solving equation of parabola with x-axis ( y=0)

We get (ab)x2+(bc)x+(ca)=0, which should have two equal values of x, as x-axis touches the parabola (bc)24(ab)(ca)=0
(b+c2a)2=02a+b+c=0ax+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)

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