Question

Find the​ point, P, at which the line intersects the plane. x equals 7 plus 9 t​, y equals 3 minus 7 t​, z equals 7 minus 5 t​; 5 x minus 6 y minus 9 z equals negative 1 \

Answer 1

The point of intersection
`P\left((1133)/(122),(149)/(122),(699)/(122)\right)`

Explanation:

Equation of line:

`x=7+9t`

`y=3-7t`

`z=7-5t`

Equation of plane:

`5x-6y-7z=-1`

We need to find the point of intersection of line and plane.

Point of intersection: When both line and plane meet at single point.

So, put the value of x, y and z into plane.

`5(7+9t)-6(3-7t)-7(7-5t)=-1`

`35+45t-18+42t-49+35t=-1`

`122t=-1+32`

`t=(31)/(122)`

Substitute the value of t into x, y and z

`x=7+9\cdot (31)/(122)=(1133)/(122)`

`y=3-7\cdot (31)/(122)=(149)/(122)`

`z=7-5\cdot (31)/(122)=(699)/(122)`

Point of intersection:

`\left((1133)/(122),(149)/(122),(699)/(122)\right)`

Hence, The point of intersection
`P\left((1133)/(122),(149)/(122),(699)/(122)\right)`