lf $p, q, r$ are distinct, then $(q - r) x + (r - p) y + (p - q) = 0$ and $(q^{3} - r^{3}) x + (r^{3} - p^{3}) y + (p^{3} - q^{3}) = 0$ represents the same line if

p + q + r = 0

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p = q = r

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p^{2} + q^{2} + r^{2} = 0

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p^{3} + q^{3} + r^{3} = 0

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Solution

The correct option is D $p + q + r = 0$
Given
$L_{1} : (q - r) x + (r - p) y + (p - q) = 0$

$L_{2} : (q^{3} - r^{3}) x + (r^{3} - p^{3}) y + (p^{3} - q^{3}) = 0$
If these two lines represent the same line, then the coefficients should be in the same ratios.
$\frac{q - r}{q^{3} - r^{3}} = \frac{r - p}{r^{3} - p^{3}} = \frac{p - q}{p^{3} - q^{3}}$

$\frac{1}{q^{2} + q r + r^{2}} = \frac{1}{r^{2} + r p + p^{2}} = \frac{1}{p^{2} + p q + q^{2}}$

$\Rightarrow q^{2} + r^{2} + q r = r^{2} + p^{2} + p r$
$\Rightarrow q^{2} - p^{2} = r (p - q)$
$\Rightarrow - (p + q) = r \Rightarrow p + q + r = 0$
Hence, option A is the correct answer.

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