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The image of the line   $\frac{x-1}{3}=\frac{y-3}{1}=\frac{z-4}{-5}$    in the plane 2x-y+z+3=0 is the line

Answer:

Explanation:

Here, the plane, line and its image are parallel to each other. So, find any point on the normal to the plane from which the image line will be passed and then find the equation of the image line.

 Here, plane and line are parallel to each other, Equation of normal to the plane through the point (1,3,4) is 

   $\frac{x-1}{2}=\frac{y-3}{-1}=\frac{z-4}{1}$ = k              [say]

 Any point in this normal is 

                         (2k+1, -k+3,4+k)

  Then   $\left(\frac{2k+1+1}{2},\frac{3-k+3}{2},\frac{4+k+4}{2}\right)$   lies on plane  $\Rightarrow 2$

  $(k+1)-\left(\frac{6-k}{2}\right)+\left(\frac{8+k}{2}\right)+3=0$ 

    $\Rightarrow$               k=- 2

 Hence, point through  which this image pass is 

       (2k+1, 3-k,4+k)

 i.e,     [2(-2)+1.3+2.4-2]=(-3.5,2)

 Hence , equation of image line is

                                   $\frac{x+3}{3}=\frac{y-5}{1}=\frac{z-2}{-5}$