Question

Find the coordinates of the point where the line through (5, 1, 6) and (3, 4, 1) crosses the YZ-plane

Answer 1

The given points are A=( 5,1,6 ) and B=( 3,4,1 ).

We know that the equation of the line passing through the points, ( x 1 , y 1 , z 1 ) and ( x 2 , y 2 , z 2 ), is given by,

x− x 1 x 2 − x 1 = y− y 1 y 2 − y 1 = z− z 1 z 2 − z 1 =k (1)

Substitute the values of x 1 , y 1 , z 1 and x 2 , y 2 , z 2 in equation (1).

x−5 3−5 = y−1 4−1 = z−6 1−6 =k x−5 −2 = y−1 3 = z−6 −5 =k

x−5=−2k x=−2k+5

y−1=3k y=3k+1

z−6=−5k z=−5k+6

So, a point on the given line is of the form, ( 5−2k,3k+1,6−5k ).

We know that, the equation of YZ –plane is x=0.

Since, the line passes through the YZ –plane,

5−2k=0 k= 5 2

3k+1=3( 5 2 )+1 = 17 2

6−5k=6−5( 5 2 ) =− 13 2

Thus, the required point is ( 0, 17 2 , −13 2 ).