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

Question

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

Smart Achievers · Accepted Answer

(1)

$Let (h, k) be the point of reflection of the given point (4, - 13)$ $about the line 5 x + y + 6 = 0 .$

$The mid-point of the line segment joining points (h, k) and (4, - 13) is given by$

$(\frac{h + 4}{2}, \frac{k - 13}{2})$

$This point will lie on the given line, so we have$

$5 (\frac{h + 4}{2}) + \frac{k - 13}{2} + 6 = 0$ or $5 h + k + 19 = 0$ ...(i)

$Again the slope of the line joining points (h, k) and (4, - 13) is given by$ $\frac{k + 13}{h - 4} .$

$This line is perpendicular to the given line, therefore$ $(- 5) (\frac{k + 13}{h - 4}) = - 1$

$This gives$ $5 k + 65 = h - 4$ $or$ $h - 5 k - 69 = 0$ ...(ii)

$On solving (i) and (ii), we get$ $h = - 1 and k = - 14 .$

$Thus the point$ $(- 1, - 14)$ $is the reflection of the given point.$

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