Find the equation of the plane which contains the line of inters

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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.

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 Multiple Choice QuestionsLong Answer Type

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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.


stack 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 0

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 equals 0

straight r with rightwards arrow on top space 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 plus straight lambda open curly brackets 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 close curly brackets minus 4 plus 5 straight lambda space equals 0

rightwards double arrow space straight r with rightwards arrow on top left square bracket left parenthesis 1 minus 2 straight lambda right parenthesis straight i with hat on top plus left parenthesis negative 2 plus straight lambda right parenthesis straight j with hat on top space plus left parenthesis 3 plus straight lambda right parenthesis straight k with hat on top right square bracket minus 4 plus 5 straight lambda equals 0

left parenthesis 1 minus 2 straight lambda right parenthesis space straight x space plus left parenthesis negative 2 plus straight lambda right parenthesis straight y plus left parenthesis 3 plus straight lambda right parenthesis straight z space equals negative 5 straight lambda plus 4

rightwards double arrow space fraction numerator straight x over denominator begin display style fraction numerator negative 5 straight lambda plus 4 over denominator 1 minus 2 straight lambda end fraction end style end fraction plus fraction numerator straight y over denominator begin display style fraction numerator negative 5 straight lambda plus 4 over denominator negative 2 plus straight lambda end fraction end style end fraction plus fraction numerator straight z over denominator begin display style fraction numerator negative 5 straight lambda plus 4 over denominator 3 plus straight lambda end fraction end style end fraction equals 1

therefore space fraction numerator negative 5 straight lambda plus 4 over denominator 1 minus 2 straight lambda end fraction space equals fraction numerator negative 5 straight lambda plus 4 over denominator negative 2 plus straight lambda end fraction

rightwards double arrow space 1 minus 2 straight lambda space equals negative 2 plus straight lambda

rightwards double arrow space minus 3 straight lambda space equals space minus 3

rightwards double arrow space straight lambda space equals 1

therefore space equation space of space the space required space plane
minus straight x minus straight y plus 4 straight z equals 1
straight x plus straight y minus 4 straight z minus 1 space equals space 0

vector space equation space of space the space required space plane

equals straight r with rightwards arrow on top. left parenthesis straight i with hat on top plus straight j with hat on top minus 4 straight k with hat on top right parenthesis minus 1 space equals 0
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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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