Question

find the image of the point (4,-13) on the line 5x + y + 6 = 0.​

Answer 1

The image of the point (4,-13) on the line $5x+y+6=0$ is (-1,-1)

Step-by-step explanation:

let the coordinate of the image point be (x,y)

so the line joining the point (4,-13) and (x,y) will be the perpendicular to the straight line $5x+y+6=0$

The slope of the line $5x+y+6=0$ is given by

         $=>y=-5x-6\\=>m=-5$

∵ the line joining the point (4,-13) and (x,y) is perpendicular to line $5x+y+6=0$

∴ the slope of line joining the point (4,-13) and (x,y) is= $\frac{1}{5}$

Equation of the line joining the points (4,-13) and (x,y) is given by

                       $\frac{y+13}{x-4} =\frac{1}{5}$

                   $x-5y=69$......eq(1)

also the midpoint of the line joining the point (4,-13) and (x,y) will lie on the straight line $5x+y+6=0$)

∴ midpoint the point (4,-13) and (x,y),

          $x1=\frac{4+x}{2},y2=\frac{y-13}{2}$

therefore we have

               $5(\frac{x+4}{2})+(\frac{y-13}{2} )+6=0\\5x+y+19=0$........eq(2)

NOW, eQ(1)+5eq(2)

$=>x-5y+25x+5y=69-95\\=>26x=-26\\=>x=-1$            

∴y=-1

Hence the image of the point (4,-13) is (-1,-1).