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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

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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

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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

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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

The equations of given planes are
4x + 8 y + z – 8 = 0    ...(1)
and 0x + y + z – 4 = 0    ...(2)
Direction ratios of normal to plane (1) are 4, 8, 1.
Direction ratios of normal to plane (2) are 0, 1, 1.
Now, $4 over 0 not equal to 8 over 1 not equal to 1 over 1$
∴ given planes (1) and (2) are not parallel.
Again (4) (0) + (8) (1) + (1) (1) = 0 + 8 + 1 = 9 ≠ 0
∴ given planes (1) and (2) are not perpendicular to each other.
Let θ be angle between planes.
$therefore space space space space cos space straight theta space equals space fraction numerator left parenthesis 4 right parenthesis thin space left parenthesis 0 right parenthesis space plus space left parenthesis 8 right parenthesis thin space left parenthesis 1 right parenthesis thin space plus space left parenthesis 1 right parenthesis thin space left parenthesis 1 right parenthesis over denominator square root of left parenthesis 4 right parenthesis squared plus left parenthesis 8 right parenthesis squared plus left parenthesis 1 right parenthesis squared end root space square root of left parenthesis 0 right parenthesis squared plus left parenthesis 1 right parenthesis squared plus left parenthesis 1 right parenthesis squared end root end fraction$
   $equals space fraction numerator 0 plus 8 plus 1 over denominator square root of 16 plus 64 plus 1 end root space square root of 0 plus 1 plus 1 end root end fraction space equals fraction numerator 9 over denominator square root of 81 space square root of 2 end fraction space equals fraction numerator 9 over denominator 9 square root of 2 end fraction space equals fraction numerator 1 over denominator square root of 2 end fraction$
$therefore space space space straight theta space equals space 45 space degree.$

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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).

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Long Answer Type

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

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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).

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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