Question

Find the coordinates of the point where the line through (3,-4,-5) and (2,-3,1) crosses the plane 2x + y + z = 7

Answer 1

Hey !!

The equation of given plane is 2x + y + z = 7 ----------> (1)

Equation of the line passing through points (3 , -4 , -5) and (2 , -3 , 1) is

             $\frac{x-3}{2-3} = \frac{y+4}{-3+4} = \frac{z+5}{1+5}$

             $\frac{x-3}{-1} = \frac{y+4}{1} = \frac{z+5}{6}$ = λ

$\frac{x-3}{-1}$ = λ     => x - 3 = -λ        => x = 3 - λ

$\frac{y+4}{1}$ = λ   => y + 4 = λ      => z = -5 + 6λ

   $\frac{z+5}{6}$ = λ        => z + 5 = 6λ         => z = -5 + 6λ

Putting the values of x, y and z in (1)

  =>  2(3 - λ) + (-4 + λ) + (-5 + 6λ) = 7

  =>   6 - 2λ- 4 + λ - 5 + 6λ = 7

  =>   5λ = 7 + 3

   => λ = 2

∴  x = 3 - 2 = 1

 y = -4 + 2 = -2

z = -5 + 6(2) = -5 + 12 = 7

Thus, coordinate of required point are ( 1 , -2 , 7)

GOOD LUCK !!