Find the equation of the plane passing through the point (1, 4, -2) and parallel to the plane 2x - y + 3z + 7 = 0.

Solution:

Given point (1, 4, -2)

Equation of the plane passing through the point (1, 4, -2) is

A(x - 1) + B(y - 4) + C(z + 2) = 0 --- (1)

Since the plane is parallel to the plane its equation is

-2x + y - 3z = 7

So, A/-2 = B/1 = C/-3 = k (k is any constant)

A = -2k,B = k,C = -3k

putting the values of A, B, C in eq(1)

-2k(x - 1) + k(y - 4) - 3k(z + 2) = 0

-2(x - 1) + y - 4 - 3(z + 2) = 0

-2x + 2 + y - 4 - 3z - 6 = 0

2x - y + 3z + 8 = 0

The equation of plane is 2x - y + 3z + 8 = 0

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Find the equation of the plane passing through the point (1, 4, -2) and parallel to the plane 2x - y + 3z + 7 = 0.

Summary:

The equation of the plane passing through the point (1, 4, -2) and parallel to the plane 2x - y + 3z + 7 = 0 is 2x - y + 3z + 8 = 0.