Find the coordinates of the foot of perpendicular drawn from the

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 Multiple Choice QuestionsLong Answer Type

301. Show that the lines:
fraction numerator straight x minus 5 over denominator 4 end fraction space equals space fraction numerator straight y minus 7 over denominator 4 end fraction space equals fraction numerator straight z plus 3 over denominator negative 5 end fraction comma space space fraction numerator straight x minus 8 over denominator 7 end fraction space equals space fraction numerator straight y minus 4 over denominator 1 end fraction space equals space fraction numerator straight z minus 5 over denominator 3 end fraction are co-planar, find their common point and the equation of the plane in which way they lie. 
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 Multiple Choice QuestionsShort Answer Type

302. Find the angle between each of the following pairs of lines:
A line with direction ratios 2 : 2 : 1.
A line joining (3, 1, 4) to (7, 2, 12)
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303. Find the angle between each of the following pairs of lines:
5, –12, 13;    – 3, 4, 5
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304. Find the angle between each of the following pairs of lines:
1, 1, 2;   square root of 3 minus 1 comma space space minus square root of 3 minus 1 comma space 4
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305.

Write the sum of intercepts cut off by the planestraight r with rightwards arrow on top. open parentheses 2 straight i with hat on top plus straight j with hat on top minus straight k with hat on top close parentheses minus 5 space equals 0 on the three axes.

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306.

Find the coordinates of the foot of perpendicular drawn from the point A
(-1,8,4) to the line joining the points B(0,-1,3) and C(2,-3,-1). Hence find the image of the point A in the line BC.


Suppose P be the foot of the perpendicular drawn from point A on the line joining
points B and C.
Let P’ (a, b, c) be the coordinates of the image of point A.
fraction numerator straight x minus straight x subscript 1 over denominator straight x subscript 2 minus straight x subscript 1 end fraction space equals space fraction numerator straight y minus straight y subscript 1 over denominator straight y subscript 2 minus end subscript straight y subscript 1 end fraction space space equals space fraction numerator straight z minus straight z subscript 1 over denominator straight z subscript 2 minus straight z subscript 1 end fraction

fraction numerator straight x minus 0 over denominator 2 end fraction space equals space fraction numerator space straight y plus 1 over denominator negative 2 end fraction space equals space fraction numerator straight z minus 3 over denominator negative 4 end fraction space equals space straight lambda

General space coordinates space of space straight P space is space left parenthesis 2 straight lambda comma negative 2 straight lambda minus 1 comma negative 4 straight lambda plus 3 right parenthesis

Direction space ratio space of space AP space left parenthesis 2 straight lambda plus 1 comma negative 2 straight lambda minus 9 comma 4 straight lambda minus 1 right parenthesis

AP perpendicular BC

2 left parenthesis 2 straight lambda plus 1 right parenthesis minus 2 left parenthesis negative 2 straight lambda minus 9 right parenthesis minus 4 left parenthesis negative 4 straight lambda minus 1 right parenthesis equals 0

4 straight lambda plus 2 plus 4 straight lambda plus 18 plus 16 straight lambda plus 4 space equals 0

24 plus 24 straight lambda space equals 0

straight lambda equals negative 1

straight P left parenthesis negative 2 comma 1 comma 7 right parenthesis

Coordinates space of space foot space of space perpendicular space is space left parenthesis negative 2 comma 1 comma 7 right parenthesis

Coordinates space of space image space of space straight A space is space straight P to the power of apostrophe left parenthesis space straight a comma straight b comma straight c right parenthesis space is space

fraction numerator straight a minus 1 over denominator 2 end fraction equals negative 2 comma space straight a equals negative 3

fraction numerator straight b plus 8 over denominator 2 end fraction space equals space 1 comma space straight b equals negative 6

fraction numerator straight c plus 4 over denominator 2 end fraction space equals space 7 comma space straight c equals space 10

straight P apostrophe space left parenthesis negative 3 comma negative 6 comma 10 right parenthesis

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 Multiple Choice QuestionsLong Answer Type

307.

Find the equation of the plane which contains the line of intersection of the planes
straight r with rightwards arrow on top. left parenthesis straight i with hat on top minus 2 straight j with hat on top plus 3 straight k with hat on top right parenthesis minus 4 space equals space 0
and
straight r with rightwards arrow on top. left parenthesis negative 2 straight i with hat on top plus straight j with hat on top plus straight k with hat on top right parenthesis plus 5 space space equals space 0

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

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 Multiple Choice QuestionsShort Answer Type

308.

Find the distance of a point (2, 5, −3) from the plane straight r with rightwards arrow on top. open parentheses 6 straight i with hat on top minus 3 straight j with hat on top plus 2 straight k with hat on top close parentheses space equals 4.

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309.

Find x such that the four points A(4, 1, 2), B(5, x, 6) , C(5, 1, -1) and D(7, 4, 0) are coplanar.

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310.

A line passing through the point A with position vector straight a with rightwards arrow on top space equals space 4 straight i with hat on top plus 2 straight j with hat on top plus 2 straight k with hat on top is parallel to the vector straight b with rightwards arrow on top space equals space 2 straight i with hat on top plus 3 straight j with hat on top plus 6 straight k with hat on top. Find the length of the perpendicular drawn on this line from a point p with vector stack straight r subscript 1 with rightwards arrow on top space equals straight i with hat on top plus 2 straight j with hat on top plus 3 straight k with hat on top.

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