Question

A focal chord

S E

of the parabola

y^{2} = 8 x

passes through the end point, having positive co-ordinates, of another chord

E E^{'}

of which the equation is

x = 4

. Find the equation and the length of the chord

S E

.

Accepted Answer

Given, equation of parabola is $y^{2} = 8 x$

Comparing with $y^{2} = 4 ax$ we get $4 a = 8 \Rightarrow a = 2$

$∴$ Focus, $S$ is $(2, 0)$

$∵$ Equation of chord ${EE}^{'}$ is $x = 4$

Put $x = 4$ in equation (1)

$\Rightarrow y^{2} = 8 \times 4$

$\Rightarrow y^{2} = 32$

$\Rightarrow y = \pm \sqrt{32} = \pm 4 \sqrt{2}$

$∴$ Co-ordinates of $E$ are $(4, 4 \sqrt{2})$

Now, by using the two point from of a line i.e. $y - y_{1} = \frac{y_{2} - y_{1}}{x_{2} - x_{1}} (x - x_{1})$

where $(x_{1}, y_{1})$ and $(x_{1}, y_{2})$ are two points which lie on the line.

So, equation of chord SE is, $y^{2} = 8 x$

$\Rightarrow y = \frac{4 \sqrt{2}}{2} (x - 2)$

$\Rightarrow y = 2 \sqrt{2} (x - 2)$

$\Rightarrow y = 2 \sqrt{2} x - 4 \sqrt{2} . . . . . . . . . . . . (2)$

To find the other end of the chord SE solve and (2)

Put the value of $y$ from (2) in (1), we get

${(2 \sqrt{2} x - 4 \sqrt{2})}^{2} = 8 x$

$\Rightarrow 8 x^{2} + 32 - 32 x = 8 x$

$\Rightarrow 8 x^{2} - 40 x + 32 = 0$

$\Rightarrow x^{2} - 5 x + 4 = 0 \Rightarrow x^{2} - 4 x - x + 4 = 0$

$\Rightarrow x = 1$ or $x = 4$

$∴ y = 2 \sqrt{2} - 4 \sqrt{2}$

$\Rightarrow y = - 2 \sqrt{2}$

So, other end of the chord SE is $(1, - 2 \sqrt{2})$

$∵$ By distance formula between two points $(x_{1}, y_{1})$ & $(x_{2}, y_{2})$ i.e. $\sqrt{{(x_{2} - x_{1})}^{2} + {(y_{2} - y_{1})}^{2}}$

$∴$ Length of chord $SE = \sqrt{{(4 - 1)}^{2} + {(4 \sqrt{2} + 2 \sqrt{2})}^{2}}$

$= \sqrt{3^{2} + {(6 \sqrt{2})}^{2}}$

$= \sqrt{9 + 36 \times 2}$

$= \sqrt{9 + 72} = \sqrt{81} = 9$ unit

A focal chord $S E$ of the parabola $y^{2} = 8 x$ passes through the end point, having positive co-ordinates, of another chord $E E^{'}$ of which the equation is $x = 4$ . Find the equation and the length of the chord $S E$ .

Important Questions on Parabola

Learn and Prepare for any exam you want !

A focal chord SE of the parabola y2=8x passes through the end point, having positive co-ordinates, of another chord EE' of which the equation is x=4. Find the equation and the length of the chord SE.

Important Questions on Parabola

Learn and Prepare for any exam you want !

A focal chord $S E$ of the parabola $y^{2} = 8 x$ passes through the end point, having positive co-ordinates, of another chord $E E^{'}$ of which the equation is $x = 4$ . Find the equation and the length of the chord $S E$ .