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Find the value of m, if the lines joining the origin and the point of intersection of y=m x+1 and x^{2}+3 y^{2}=1 are perpendicular to one another

Option: 1

\sqrt{3}


Option: 2

-\sqrt{3}


Option: 3

A and B both


Option: 4

None of these


Answers (1)

best_answer

\\\text{Given curve is }x^{2}+3 y^{2}=1 .. (i)\text{ and given line is }y=m x+1 ...(ii)

Homogenizing the curve to get equation of pair of lines passing through their point of intersection

x^{2}+3 y^{2}=(y-m x)^{2} \\ x^{2}+3 y^{2}=y^{2}+m^{2} x^{2}-2 m x y \\ \left(1-m^{2}\right) x^{2}+2 m x y+2 y^{2}=0\\

For these lines to be perpendicular

\tan(90^o) = \frac {2\sqrt{h^2-ab}}{|a+b|}

So, a + b = 0

\\Hence,(1-m^2)+2=0\\ m=\pm \sqrt{3}

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