If the parabola y = ( a − b ) x 2 + ( b − c ) x + ( c − a ) touches the x -axis then the line a x + b y + c = 0

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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 $\Rightarrow (b - c)^{2} - 4 (a - b) (c - a) = 0$ $\Rightarrow (b + c - 2 a)^{2} = 0 \Rightarrow - 2 a + b + c = 0 \Rightarrow a x + b y + c = 0 always passes through (- 2, 1)$

Mathematics - Parabola Question with Solution | TestHub

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