The points of intersection of the parabolas $y^{2} = 5 x$ and $x^{2} = 5 y$ lie on the line

x + y = 10

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x - 2 y = 0

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x - y = 0

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2 x - y = 0

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Solution

The correct option is C $x - y = 0$
The point of intersection of parabolas $y^{2} = 5 x$ and $x^{2} = 5 y$ is given by ${(\frac{x^{2}}{5})}^{2} = 5 x$
$\Rightarrow x^{4} = 25 \times 5 x$
$\Rightarrow x^{4} = 125 x$
$∴ x (x^{3} - 125) = 0$
$∴ x (x - 5) (x^{2} + 5 x + 25) = 0$
The real values of $x$ satisfying the above equation are $x = 0, 5$
Corresponding points of intersection on the parabolas are $(0, 0)$ and $(5, 5)$
The equation of line passing through these points is $y - x = 0$ i.e. $x - y = 0$

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The points of intersection of the parabolas y2=5x and x2=5y lie on the line

The points of intersection of the parabolas $y^{2} = 5 x$ and $x^{2} = 5 y$ lie on the line