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  • A line through the origin intersects <img alt="\mathrm{x=1, y=2 \; and\; x+y=4}" src="https://entrancecorner.oncodecogs.com/gif.latex?%5Cmathrm%7Bx%3D1%2C%20y%3D2%20%5C%3B%20and%5C%3B%20x&plus;

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

Option: 1

\mathrm{y=x}


Option: 2

\mathrm{y=-x}


Option: 3

\mathrm{y=2x}


Option: 4

\mathrm{y=x+1}


Answers (1)

best_answer

Let the equation of the line through the origin be \mathrm{\frac{\mathrm{x}}{\cos \theta}=\frac{\mathrm{y}}{\sin \theta}=\mathrm{r} \text {. }} Any point on this line is \mathrm{(r \ cos \: \theta, r \sin\:\theta)}. The combined equation of given three lines is \mathrm{(x-1)(y-2)(x+y-4)=0}. Now OA, OB and OC are the roots of \mathrm{(r \cos \theta-1)} \mathrm{(r \sin \theta-2)(r \cos \theta+r \sin \theta-4)=0}

Thus \mathrm{\text { OA.OB.OC }=\frac{8}{\sin \theta \cos \theta(\sin \theta+\cos \theta)}=8 \sqrt{2}} (given)

\begin{aligned} & \mathrm{\Rightarrow \sin \theta \cos \theta(\sin \theta+\cos \theta)=\frac{1}{\sqrt{2}} \Rightarrow \sin 2 \theta(\sin \theta+\cos \theta)=\sqrt{2}} \\ \\& \mathrm{ \Rightarrow \sin 2 \theta \cdot \sin (\theta+\pi / 4)=1 \Rightarrow \sin 2 \theta=1, \sin (\theta+\pi / 4)=1 \Rightarrow \theta=\pi / 4} . \end{aligned}

Thus the required equation of the line is y = x.

Posted by

Pankaj

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