Question

The reflection of the point $(4,-13)$ about the line $5x+y+6=0$ is

• A. $(-1,-14)$
• B. $(3,4)$
• C. $(0,0)$
• D. $(1,2)$

Answer 1

Answer: (A)

Let $(h,k)$ be the point of the given point $(4,-13)$ about the line $5x+y+6=0$. The mid-point of the line segment joining points $(h,k)$ and $(4,-13)$ is given by
$\left(\frac{h+4}{2},\frac{k-13}{2}\right)$
This point lies on the given line, so we have
$5\left(\frac{h+4}{2}\right)+\frac{k-13}{2}+6=0$
or $5h+k+19=\mathrm{0.....}$(i)
Again, the slope of the line joining points $(h,k)$ and $(4,-13)$ is given by $\frac{k+13}{h-4}$. This line is perpendicular to the given line and hence $(-5)\left(\frac{k+13}{h-4}\right)=-1$
This gives $5k+65=h-4$
or $h-5k-69=\mathrm{0.....}$(ii)
On solving Eqs. (i) and (ii), we get
$h=-1\text{ and }k=-14$
Thus, the point $(-1,-14)$ is the reflection of the given point.