Find the length of chord cut by line $4 y - 8$ at parabola $y^{2} = 8 x$ .

Open in App

Solution

Equation of parabola,
$y^{2} = 8 x . . . (i)$
Equation of line,
$4 y - 3 x = 8$
$\Rightarrow 4 y = 3 x + 8$
$\Rightarrow y = \frac{3 x + 8}{4} . . . . (i i)$
From equation (i) and (ii)
${(\frac{3 x + 8}{4})}^{2} = 8 x$
$\Rightarrow (3 x + 8)^{2} = 128 x$
$\Rightarrow 9 x^{2} + 48 x + 64 - 128 x = 0$
$\Rightarrow 9 x^{2} - 80 x + 64 = 0.... (i i i)$
On solving,
$x = - (- 80) \pm \frac{\sqrt{(- 80)^{2} - 4 \times 9 \times 64}}{2 \times 9}$
$= \frac{80 \pm \sqrt{6400 - 2304}}{18} = \frac{80 \pm \sqrt{4096}}{18}$
$= \frac{80 \pm 64}{18} = \frac{40 \pm 32}{9} = 8$
Thus $x = 8, \frac{8}{9}$
Put value of $x$ in eqn (i)
$y^{2} = 8 \times 8 \Rightarrow y = 8$
$y^{2} = 8 \times \frac{8}{9} \Rightarrow y = \frac{8}{3}$
Thus, coordinates of ends of chord are $(8, 8)$ and $(\frac{8}{9}, \frac{8}{3})$ . Thus length of chord
$= \sqrt{{(8 - \frac{8}{9})}^{2} + {(8 - \frac{8}{3})}^{2}}$
$= \sqrt{{(\frac{64}{9})}^{2} + {(\frac{16}{3})}^{2}} = \sqrt{\frac{4096}{81} + \frac{256}{9}}$
$= \sqrt{\frac{4096 + 2304}{81}} = \sqrt{\frac{6400}{81}} = \frac{80}{9}$
Thus, length of chord is $\frac{80}{9}$ units.

Suggest Corrections

Similar questions

y^{2} = 8 x

x + y = 8

x^{2} = 4 y

x^{2} + y^{2} - 8 x - 2 y - 8 = 0

x + y + 1 = 0

y^{2} = 8 x

x + y = 1

Whats the length of chord intercepted by the parabola $y^{2} = 4 x o n t h e l i n e x + y = 0$

Join BYJU'S Learning Program