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Equation of Normal at Given Point

Show that the...

Question

Show that the line $x + y = 4$ intersects the parabola $y^{2} = 2 x$ at two. Find the coordinates of these points and the length $P Q$ .

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Solution

$\begin{matrix} x + y = 4 \to (i) y^{2} = 2 x \to (i i) f r o m e q . (i) v a l u e o f n = 4 - y p u t t i n g i n e q . (i i) y^{2} = 2 (4 - y) y^{2} = 8 - 2 y y^{2} + 2 y - 8 = 0 y^{2} + 4 y - 2 y - 8 = 0 y (y + 4) - 2 (y + 4) = 0 (y - 2) (y + 4) = 0 y = 2, - 4 i f, y = 2 t h e n x = 2 i f y = - 4 t h e n, x = 8 P Q = \sqrt{{(2 - 8)}^{2} + {(2 + 4)}^{2}} P Q = \sqrt{36 + 36} = P Q = 6 \sqrt{2} u n i t s \end{matrix}$

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