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Question: Drawn from the origin are two mutually perpendicular lines forming an isosceles triangle together wi...

Drawn from the origin are two mutually perpendicular lines forming an isosceles triangle together with the straight line 2x+y = a. Then the area of this triangle is

A

a22\frac { a ^ { 2 } } { 2 }

B

a23\frac { a ^ { 2 } } { 3 }

C

a25\frac { a ^ { 2 } } { 5 }

D

None of these

Answer

a25\frac { a ^ { 2 } } { 5 }

Explanation

Solution

Let the two perpendicular lines through the origin intersect

2x+y = a at A and B so that the triangle OAB is isosceles.

OM = length of perpendicular from O to AB; OM =.

Also AM = MB = OM ⇒ AB =.

Area of ∆ OAB = 12\frac { 1 } { 2 }AB. OM = 12⋅2a5⋅a5=a25\frac { 1 } { 2 } \cdot \frac { 2 a } { \sqrt { 5 } } \cdot \frac { a } { \sqrt { 5 } } = \frac { a ^ { 2 } } { 5 }