Straight Line Question 124

Question: Coordinates of the foot of the perpendicular drawn from (0,0) to the line joining $ (a\cos \alpha ,a\sin \alpha ) $ and $ (a\cos \beta ,a\sin \beta ) $ are [IIT 1982]

Options:

A) $ ( \frac{a}{2},\frac{b}{2} ) $

B) $ [ \frac{a}{2}(\cos \alpha +\cos \beta ),\frac{a}{2}(\sin \alpha +\sin \beta ) ] $

C) $ ( \cos \frac{\alpha +\beta }{2},\sin \frac{\alpha +\beta }{2} ) $

D)None of these

Show Answer

Answer:

Correct Answer: B

Solution:

  • Slope of perpendicular = ? $ [ \frac{\cos \alpha -\cos \beta }{\sin \alpha -\sin \beta } ] $ $ =-\cot \frac{\alpha +\beta }{2} $ Hence equation of perpendicular is $ y=-\cot ( \frac{\alpha +\beta }{2} )\text{ }x $?.. (i) Now on solving the equation (i) with the line, we get the required point.Trick: Take suitable values of $ a,\alpha ,\beta $ and then check with options.



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