If the perpendicular bisector of the line
Question:

If the perpendicular bisector of the line segment joining the points $P(1,4)$ and $Q(k, 3)$ has $y$-intercept equal to $-4$, then a value of $k$ is :

  1. (1) $-2$

  2. (2) $-4$

  3. (3) $\sqrt{14}$

  4. (4) $\sqrt{15}$


Correct Option: , 2

Solution:

Mid point of line segment $P Q$ be $\left(\frac{k+1}{2}, \frac{7}{2}\right)$.

$\therefore$ Slope of perpendicular line passing through

$(0,-4)$ and $\left(\frac{k+1}{2}, \frac{7}{2}\right)=\frac{\frac{7}{2}+4}{\frac{k+1}{2}-0}=\frac{15}{k+1}$

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

$\therefore \frac{15}{1+k} \times \frac{1}{1-k}=-1$

$1-k^{2}=-15 \Rightarrow k=\pm 4$

 

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