The Line 2x − Y + 4 = 0 Cuts the Parabola Y2 = 8x in P and Q. the Mid-point of Pq is - Mathematics

MCQ

The line 2x − y + 4 = 0 cuts the parabola y2 = 8x in P and Q. The mid-point of PQ is

•  (1, 2)

•  (1, −2)

•  (−1, 2)

•  (−1, −2)

Solution

(−1, 2)

Let the coordinates of P and Q be $\left( a {t_1}^2 , 2a t_1 \right)$ and $\left( a {t_2}^2 , 2a t_2 \right)$  respectively.
Slope of PQ = $\frac{2a t_2 - 2a t_1}{a {t_2}^2 - a {t_1}^2}$             ......(1)

But, the slope of PQ is equal to the slope of 2x − y + 4 = 0.
∴ Slope of PQ = $\frac{- 2}{- 1} = 2$

From (1), $\frac{2a t_2 - 2a t_1}{a {t_2}^2 - a {t_1}^2} = 2$                        .....(2)

Putting 4a = 8,
a = 2

∴ Focus of the given parabola = (a, 0) = $\left( 2, 0 \right)$

Using equation (2):

$\frac{4\left( t_2 - t_1 \right)}{2\left( {t_2}^2 - {t_1}^2 \right)} = 2$

$\frac{\left( t_2 - t_1 \right)}{\left( {t_2}^2 - {t_1}^2 \right)} = 1$

$\frac{\left( t_2 - t_1 \right)}{\left( {t_2}^2 - {t_1}^2 \right)} = 1$

As, points P and Q lie on 2x-y+4=0

$\Rightarrow P(a {t_1}^2 , 2a t_1 ) or P(2 {t_1}^2 , 4 t_1 ) \text{ lie on line } 2x - y + 4 = 0$
$\Rightarrow 2\left( 2 {t_1}^2 \right) - \left( 4 t_1 \right) + 4 = 0$
$\Rightarrow {t_1}^2 - t_1 + 1 = 0 . . . (3)$
$\text{ Also }, Q(a {t_2}^2 , 2a t_2 ) or P(2 {t_2}^2 , 4 t_2 ) \text{ lie on line } 2x - y + 4 = 0$
$\Rightarrow 2\left( 2 {t_2}^2 \right) - \left( 4 t_2 \right) + 4 = 0$
$\Rightarrow {t_2}^2 - t_2 + 1 = 0 . . . (4)$
$\text{ Adding } (3) \text{ and } (4), \text{ we get },$
$\Rightarrow {t_1}^2 - t_1 + 1 + {t_2}^2 - t_2 + 1 = 0$
$\Rightarrow \left( {t_1}^2 + {t_2}^2 \right) - \left( t_1 + t_2 \right) + 2 = 0$
$\Rightarrow \left( {t_1}^2 + {t_2}^2 \right) - 1 + 2 = 0 \left[ t_1 + t_2 = 1, \text{ proved above } \right]$
$\Rightarrow \left( {t_1}^2 + {t_2}^2 \right) = - 1$

Let $\left( x_1 , y_1 \right)$  be the mid-point of PQ.
Then, we have: $y_1 = \frac{2a t_2 + 2a t_1}{2} = 2\left( t_1 + t_2 \right) = 2$

$x_1 = \frac{a {t_1}^2 + a {t_2}^2}{2} = {t_1}^2 + {t_2}^2 = - 1$

⇒ $\left( x_1 , y_1 \right) = \left( - 1, 2 \right)$

Is there an error in this question or solution?

APPEARS IN

RD Sharma Class 11 Mathematics Textbook
Chapter 25 Parabola
Q 12 | Page 29