Q. 104.0( 4 Votes )

# 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.

Answer :

Let a_{1},b_{1} and c_{1} 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 a_{1}, b_{1} and c_{1} and a_{2} , b_{2} and c_{2} then

a_{1}a_{2} + b_{1}b_{2} + c_{1}c_{2} = 0

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

5a_{1} + 2b_{1} – 4c_{1} = 0 ……(1)

2a_{1} + 8b_{1} + 2c_{1} = 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 a_{2}x + b_{2}y + c_{2}z + d_{2} = 0 if a_{1}a_{2} + b_{1}b_{2} + c_{1}c_{2} = 0 ……(3)

Here, line with direction ratios a_{1},b_{1} and c_{1} 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 equation of the plane which contains the line of intersection of the planes

and

and whose intercept on the x-axis is equal to that of on y-axis.

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