If $(p^{2} - q^{2}) x^{2} + (q^{2} - r^{2}) x + r^{2} - p^{2} = 0$ and $(p^{2} - q^{2}) y^{2} + (r^{2} - p^{2}) y + q^{2} - r^{2} = 0$ have a common root for all real values of p, q and r, then find the common root.

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Solution

$(p^{2} - q^{2}) x^{2} + (q^{2} - r^{2}) x + r^{2} - p^{2} = 0$ ....(1)
The sum of the coefficients $(p^{2} - q^{2}) + (q^{2} - r^{2}) + (r^{2} - p^{2}) = 0$
Therefore, $x = 1$ is a root of equation 1.
$(p^{2} - q^{2}) y^{2} + (r^{2} - p^{2}) y + q^{2} - r^{2} = 0$ .....(2)
$p^{2} - q^{2} + r^{2} - p^{2} + q^{2} - r^{2} = 0$
Therefore, $y = 1$ is a root of equation 2.
Therefore, 1 is the common root of equations 1 and 2.

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