Question

The perpendicular bisector of the line segment joining $P(1,4)$ and $Q(k,3)$ has $y-$intercept $-4.$ Then, the possible value of $k$ is

• A. $-4$
• B. $1$
• C. $2$
• D. $-2$

Answer 1

The correct option is A

$-4$

Explanation for the correct option:

Compute the value of the given variable.

Slope of $PQ=\frac{3-4}{k-1}$

The slope of the perpendicular bisector of $PQ=(k-1)$
Also midpoint of $PQ=\left(\frac{k+1}{2},\frac{7}{2}\right)$

Equation of perpendicular bisector of $PQ$ is $y-\frac{7}{2}=(k-1)(x-\frac{k+1}{2})$
$$\begin{gathered} \Rightarrow 2y-7=2x(k-1)-(k^2-1) \\ \Rightarrow 2(k-1)x-2y+(8-k^2)=0 \end{gathered}$$

Now, $Y-$intercept is given as,

$$\begin{gathered} \frac{8-k^2}{2}=-4 \\ \Rightarrow k=\pm 4 \end{gathered}$$

Hence, correct option is $\left(\mathrm{A}\right)$