Let the line x + y = 1 meet the axes of x and y at A and B, respectively.

Question

Let the line x + y = 1 meet the axes of x and y at A and B, respectively. A right angled triangle AMN is inscribed in the triangle OAB, where O is the origin and the points M and N lie on the lines OB and AB, respectively. If the area of th...

Accepted Answer

Area of $	riangleAOB=\frac{1}{2}$

Area of $	riangleAMN=\frac{4}{9}	imes \frac{1}{2}=\frac{2}{9}$

Equation of AB is $x+y=1$

$\begin{gathered} OA=1,AM=\sec ({45}^{\circ }-	heta ) \ AN=\sec ({45}^{\circ }-	heta )\cos 	heta \ MN=\sec ({45}^{\circ }-	heta )\sin 	heta \end{gathered}$

$\begin{gathered} Ar(	riangleAMN)=\frac{1}{2}	imes {\sec }^{2}({45}^{\circ }-	heta )\sin 	heta \cdot \cos 	heta =\frac{2}{9} \ \Rightarrow 	an 	heta =2,\frac{1}{2} \ 	an 	heta =2 \ \frac{AN}{NB}=\frac{\lambda }{1}=\cot 	heta =2 \end{gathered}$

Solutions · 1