Question

Find the point, P, at which the line intersects the plane given by 9x−7y−2z=6. The line is defined by the parametric equations: x=9+7t, y=4+5t, z=−10+9t

Answer 1

To solve for the point P where the line intersects the plane, substitute the line's parametric equations into the plane's equation, solve for t, and then use this value in the parametric equations to find P's coordinates.

Step-by-step explanation:

To find the point P at which the given line intersects the plane described by the equation 9x−7y−2z=6, we need to substitute the parametric equations of the line, x=9+7t, y=4+5t, z=−10+9t, into the plane's equation and solve for the parameter t.

Substituting the parametric equations into the plane's equation gives us:
9(9+7t) - 7(4+5t) - 2(-10+9t) = 6

Simplifying, we get a quadratic equation in terms of t that we need to solve. We rearrange the equation to get 0 on one side of the equation. This equation would be of the form at² + bt + c = 0. We can then use the quadratic formula, t = (-b ± √(b² - 4ac)) / (2a), to solve for t.

Once we have the value of t, we substitute it back into the original parametric equations to find the coordinates of point P.