Question

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

• A. $(-1,-14)$
• B. $(3,4)$
• C. $(1,2)$
• D. none of these

Answer 1

The correct option is A $(-1,-14)$

$5x+y+6=\mathrm{0......}\left(1\right)$

Take a circle of zero radius at (4,-13)

$(x-4{)}^2+(y-13{)}^2=\mathrm{0.....}\left(2\right)$

With radical axis as (1) find a co axial circle

$(x-4{)}^2+(y-13{)}^2–2\lambda (5x+y+6)=\mathrm{0......}\left(3\right)$

We can resolve $\lambda$ by 2 means.

(A) The circle (3) has 0 radius.

(B) The center of (2) and (3) have opposite powers w.r.t. the line (1)

(A) involves more calculations than (B). So we adopt (B)

Center of (3)=$(5\lambda +4,\lambda -13)$

applying (B)

$5(5\lambda +4)+(\lambda +3)+6=-(5\times 4–13+6)\Rightarrow$

$\lambda =-1\lambda =-1$

So the image point required is $(5\times -1+4,-1–13)=(-1,-14)$