# Prove that the line of section of the planes 5x + 2y – 4z + 2 = 0 and 2x + 8y + 2z – 1 = 0 parallel to the plane 4x – 2y – 5z – 2 = 0.

Let a1,b1 and c1 be the direction ratios of the line 5x + 2y – 4z + 2 = 0 and 2x + 8y + 2z – 1 = 0.

As we know that if two planes are perpendicular with direction ratios as a1, b1 and c1 and a2 , b2 and c2 then

a1a2 + b1b2 + c1c2 = 0

Since, line lies in both the planes, so it is perpendicular to both planes

5a1 + 2b1 – 4c1 = 0 ……(1)

2a1 + 8b1 + 2c1 = 0 ……(2)

Solving equation (1) and (2) by cross multiplication we have,    a = 2k, b = – k and c = 2k

We know that line is parallel to plane a2x + b2y + c2z + d2 = 0 if a1a2 + b1b2 + c1c2 = 0 ……(3)

Here, line with direction ratios a1,b1 and c1 is parallel to plane,

4x – 2y – 5z = 5

So,

2×4 + (– 1)× – 2 + 2× – 5 = 0

8 + 2 – 10 = 0

Therefore, the line of section is parallel to the plane.

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Find the coordinate of the point P where the line through and crosses the plane passing through three points and Also, find the ratio in which P divides the line segment AB.

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