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Question

The line x+65=y+103=z+148 is the hypotenuse of an isosceles right-angled triangle whose opposite vertex is (7,2,4). Then which of the following is not the side of the triangle?

A
x72=y23=z46
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B
x73=y26=z42
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C
x73=y25=z41
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D
none of these
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Solution

The correct option is C x73=y25=z41
Let opposite vertex be A(7,2,4).
x+65=y+103=z+148=λ

Hence, any point on above line be B(5λ6,3λ10,8λ14)

Direction ratios of line AB are (5λ13,3λ12,8λ18)
Direction ratios of line BC are (5,3,8).

Since, the angle between AB and BC is π/4.
cosπ4=(5λ3)5+3(3λ12)+8(8λ18)52+32+82.(5λ13)2+(3λ12)2+(8λ18)2

Squaring and solving, we get λ=3,2
Hence, equation of lines are x72=y23=z46 and x73=y26=z42

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