Q. 74.0( 2 Votes )

# Find the equation of the plane through (2, 3, – 4) and (1, – 1, 3) and parallel to the x – axis.

Answer :

We know that the equation of plane passing through (x_{1},y_{1},z_{1}) is given by

a(x – x_{1}) + b(y – y_{1}) + c(z – z_{1}) = 0 ……(1)

So, equation of plane passing through (2,3, – 4) is

a(x – 2) + b(y – 3) + c(z + 4) = 0 ……(2)

It also passes through (1, – 1, – 3)

So, equation (2) must satisfy the point (1, – 1, – 3)

∴ a(1 – 2) + b(– 1 – 3) + c(– 3 + 4) = 0

⇒ – a – 4b + c = 0

⇒ a + 4b – 7c = 0 ……(3)

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 ……(4)

Here, equation(2) is parallel to x axis,

……(5)

Using (2) and (5) in equation (4)

a×1 + b×0 + c×0 = 0

⇒ a = 0

Putting the value of a in equation (3)

a – 4b + 7c = 0

⇒ 0 – 4b + 7c = 0

⇒ – 4b = – 7c

⇒ b =

Now, putting the value of a and b in equation (2)

a(x – 2) + b(y – 3) + c(z + 4)

⇒ 0(x – 2) + (y – 3) + c(z + 4) = 0

⇒

⇒ 7cy – 21c + 4cz + 16c = 0

Dividing by c we have,

7y – 21 + 4z + 16 = 0

⇒ 7y + 4z – 5 = 0

Equation of required plane is 7y + 4z – 5 = 0

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Find the equation of the plane which contains the line of intersection of the planes

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

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