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., $y{y}_1-2(x+x_1)=y_1^2-4x_1$
i.e., $y{y}_1-2x=y_1^2-2x_1$
i.e., $-2x+y{y}_1=y_1^2-2x_1$
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 $\left(\frac{5}{2},-1\right)$