Find the length of the chord 4y=3x+8 intercepted by the parabola $y^{2} = 8 x$

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Solution

The ordinates of point of intersection of the line $4 y = 3 x + 8$ and parabola $y^{2} = 8 x$ are wrt of equation
$y^{2} = 8 (\frac{4 y - 8}{3})$ [On eliminating $x$ between $4 y = 3 x + 8$ and $y^{2} = 8 x$ ]
$\Rightarrow {3 y}^{2} - 32 y + 64 = 0$
$\Rightarrow (3 y - 8) (y - 8) = 0$
$y = \frac{8}{3}$ or $y = 8$
Putting $y = \frac{8}{3}$ and $y = 8$ in $4 y = 3 x + 8$ successively we get $x = \frac{8}{9}$ and $x = 8$
The line $4 y = 3 x + 8$ and parabola $y^{2} = 8 x$ at $P (\frac{8}{9}, \frac{8}{3})$ and $Q (8, 8)$ length of chord $P Q = \sqrt{{(8 - \frac{8}{9})}^{2} + {(8 - \frac{8}{3})}^{2}} = \frac{80}{9}$ .

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