Question

The mid-point of the chord $2x + y - 4 =0$ of the parabola $y^2 = 4x$ is

• A. $\left(\frac{5}{2},-2\right)$
• B. $\left(\frac{5}{2},-1\right)$
• C. $\left(\frac{3}{2},-1\right)$
• D. none of these.

Answer 1

Answer: (B)

Let $(x_1, y_1) $ is the mid. pt. of the chord $2x + y - 4 = 0 $
Equation of the chord in terms of mid point is $T = S_1$
i.e., $yy_1 - 2 (x + x_1) = y_1^{2} - 4x_1$
i.e., $yy_1 - 2x = y_1^2 - 2x_1$
i.e., $- 2x + yy_1 =y_1^2-2x_1$
Also, chord is $2x + y = 4$
$\therefore \frac{-2}{2} = \frac{y_{1}}{1} =\frac{ y_{1}-2x_{1}}{4} $
$ \therefore y_{1} = -1$. Also, $y_{1}^{2}-2x_{1} = -4 $
$\therefore 1-2x_{1} = -4 $
$ \Rightarrow 2x_{1} = 5 $
$ \Rightarrow x_{1} = \frac{5}{2}$
$ \therefore $ reqd. mid point is $\left(\frac{5}{2}, -1\right)$