Question

# If the angle between two intersecting lines having direction ratios (5, 7, 3) & (3, 4, 5) respectively can be given by cos−1(58√b), then what will be the value of b ?

A
4150
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B
4052
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C
3971
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D
4167
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Solution

## The correct option is D 4150We know that If two lines with direction cosines l1,m1,n1 and l2,m2,n2 intersect then the angle between them would be cos−1(l1.l2+m1.m2+n1.n2) Here we are given direction ratios and not the direction cosines. So we’ll find direction cosines first. Direction cosines for the first line will be - 5√52+72+32,7√52+72+32,3√52+72+32 Or, 5√83,7√83,3√83 Similarly, direction cosines for the second line will be - 3√32+42+52,4√32+42+52,5√32+42+52 Or, 3√50,4√50,5√50 Now we can calculate the angle between these lines, which will be - cos−1(5√83.3√50+7√83.4√50+3√83.5√50) Or, cos−1(58√4150) On comparing it with the given expression we get b = 4150

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