Find an equation of the plane orthogonal to the line
(x,y,z)=(-4,-9,9)+t(-8,-1,5)
which passes through the point
(-9,9,-4).
Give your answer in the form ax+by+cz=d.
To find an equation of the plane orthogonal to a line, we need to find the normal vector of the plane.
In this case, the line is defined by the parametric equations:
x = -4 - 8t
y = -9 - t
z = 9 + 5t
To find the direction vector of the line, we take the coefficients of t in each equation:
Direction vector = (-8, -1, 5)
Since the plane is orthogonal to the line, its normal vector will be perpendicular to the direction vector of the line. Therefore, the dot product of the normal vector and the direction vector should be zero.
So, let's find the dot product of the normal vector (a, b, c) and the direction vector (-8, -1, 5):
a(-8) + b(-1) + c(5) = 0
Simplifying this equation gives us:
-8a - b + 5c = 0 ----(1)
Now, since the plane passes through the point (-9, 9, -4), we know that this point satisfies the equation of the plane. So, we can substitute the coordinates of the point (-9, 9, -4) into the equation of the plane.
ax + by + cz = d
a(-9) + b(9) + c(-4) = d
Simplifying this equation gives us:
-9a + 9b - 4c = d ----(2)
Combining equations (1) and (2), we have a system of linear equations:
-8a - b + 5c = 0 ----(1)
-9a + 9b - 4c = d ----(2)
To find the equation of the plane in the required form (ax + by + cz = d), we need to solve this system of equations.
To solve this system of equations, we can use a variety of methods such as substitution, elimination, or matrix methods.
Let's solve this system by elimination:
Multiply equation (1) by 9 and equation (2) by 8 to eliminate the b term:
-72a - 9b + 45c = 0
-72a + 72b - 32c = 8d
Then, subtract the two equations:
-9b - 72b + 45c - 72b + 72b - 32c = 0 - 8d
-109b + 13c = -8d ----(3)
Now, we have two equations: (1) and (3).
Solve the system of equations (1) and (3) simultaneously to find the values of a, b, c, and d.
After obtaining the values of a, b, c, and d, the equation of the plane in the required form (ax + by + cz = d) will be found.