Find the equation of the plane passing through the point (0, 7, -7) and containing the line (x + 1)/-3 = (y - 3)/2 = (z + 2)/1.

Solution:

We know that

Any plane passing through (0, 7, -7) is

a (x - 0) + b(y - 7) + c(z + 7) = 0 --- (i)

If (i) has the given line, it must pass through the point (-1, 3, -2) and should be parallel to the given line.

When (i) passes through (-1, 3, -2)

a(-1 - 0) + b(3 - 7) + c(-2 + 7) = 0

a + 4b - 5c = 0 --- (ii)

When (i) is parallel to the given line

(-3)a + 2b + 1.c = 0

-3a + 2b + c = 0 --- (iii)

Cross multiplying (ii) and (iii)

a/(4 + 10) = b/(15 - 1) = c/(2 + 12)

a/14 = b/14 = c/14

a/1 = b/1 = c/1 = k

a = k, b = k, c = k

Substituting a = k, b = k and c = k in (i), the required equation of the plane is k(x - 1) + k(y - 7) + k(z - 7) = 0

x + (y - 7) + (z + 7) = 0

x + y + z = 0

Therefore, the equation of the plane is x + y + z = 0.

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Find the equation of the plane passing through the point (0, 7, -7) and containing the line (x + 1)/-3 = (y - 3)/2 = (z + 2)/1.

Summary:

The equation of the plane passing through the point (0, 7, -7) and containing the line (x + 1)/-3 = (y - 3)/2 = (z + 2)/1 is x + y + z = 0.