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Three Dimensional Geometry

Short Answer Type

191. In the following cases, determine whether the given planes are parallel or perpendicular, and in case they are neither, find the angles between them:
2x + y + 3z – 2 = 0 and x – 2y + 5 = 0

The equations of given planes are
2x + y + 3z – 2 = 0 ...(1)
and x – 2y + 5 = 0 ...(2)
Direction ratios of normal to plane (1) are 2, 1, 3
Direction ratios of normal to plane (2) are 1, – 2, 0.
Now, $2 over 1 not equal to fraction numerator 1 over denominator negative 2 end fraction not equal to 3 over 0$
∴ planes (1) and (2) are not parallel.
Again (2) (1) + (1) (– 2) + (3) (0) = 2 – 2 + 0 = 0
∴ given planes (1) and (2) are perpendicular to each other.

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192. In the following cases, determine whether the given planes are parallel or perpendicular, and in case they are neither, find the angles between them:
2x – 2y + 4z + 5 = 0 and 3x – 3y + 6z – 1 = 0

85 Views

193. In the following cases, determine whether the given planes are parallel or perpendicular, and in case they are neither, find the angles between them:
2x – y + 3z – 1 = 0 and 2x – y + 3z + 3 = 0

91 Views

194. In the following cases, determine whether the given planes are parallel or perpendicular, and in case they are neither, find the angles between them:
4x + 8y + z – 8 = 0 and y + z – 4 = 0

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195. Find the equation of the plane which bisects the line joining the points (–1, 2, 3) and (3, – 5 6) at right angles.

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196. Find the equation of the plane through the intersection of the planes x + y + z = 9 and 2 x + 3 y + 4 z + 5 = 0 and passing through the point (1, 1, 1).

87 Views

Long Answer Type

197. Find the equation of plane passing through origin and intersection of planes 2x – 3y + z = 9, x – y + z = 4

93 Views

Short Answer Type

198. Find the equation of the plane through the intersection of the planes 3x – y + 2 z – 4 = 0 and x + y + z – 2 = 0 and the point (2, 2, 1).

93 Views

199. Find the equation of the plane passing through the point (– 1, – 1, 2) and perpendicular to each of the following planes:
2x + 3y – 3 = 2 and 5x – 4y + z = 6.

99 Views

Long Answer Type

200. Find the direction ratios of the normal to the plane passing through the point (2,1, 3) and the line of intersection of the planes x + 2 y + z = 3 and 2 x – y – z = 5.

334 Views