Question

A line through the origin intersects x=1,y=2 and x+y=4, at A, B, and C respectively, such that OA.OB.OC=$8\sqrt{2}$. Find the equation of the line.

• A. 2y=x
• B. y+x=0
• C. y=2x
• D. none of these

Answer 1

Answer: (A)

2y=x

Using parametric equation of line $\frac{x-0}{\cos \theta }=\frac{y-0}{\sin \theta }=r⟶$ Distance from origin
For $OA\equiv (1,m)\frac{x-0}{\cos \theta }=OA=\frac{1-0}{\cos \theta }=\frac{1}{\cos \theta }$
For $OBB(\frac{2}{m},2)\frac{y-0}{\sin \theta }=OB=\frac{2}{\sin \theta }$
For $OCC(\frac{4}{m+1},\frac{4m}{m+1})\frac{x-0}{\cos \theta }=OC=\frac{\frac{4}{m+1}}{\cos \theta }=\frac{4}{\sin \theta +\cos \theta }m=\tan \theta$
A.T.P.
$OA.OB.OC.=8\sqrt{2}\frac{1}{\cos \theta }\times \frac{2}{\sin \theta }\times \frac{4}{\sin \theta +\cos \theta }=8\sqrt{2}\frac{1}{\sqrt{2}}=(\sin \theta +\cos \theta )\sin \theta \cos \theta \theta ={45}^o$ satisfies above equation
$(y-0)=\frac{1}{2}(x-0)2y=x$