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  • Find the equation to the pair of straight lines joining the origin to the intersection points of the straight line  <img alt="\mathrm{y+m x=c}" src="https://entrancecorner.oncodecogs.com/

Find the equation to the pair of straight lines joining the origin to the intersection points of the straight line  \mathrm{y+m x=c}  and the curve \mathrm{ x^2+y^2=a^2}. They are at right angles if  \mathrm{ \mathrm{kc}^2=a^2\left(1+m^2\right)}  , where \mathrm{ k=}

Option: 1

1


Option: 2

2


Option: 3

3


Option: 4

4


Answers (1)

best_answer

Re-writing the equation of the given line \mathrm{y+m x=c} as \mathrm{ \frac{y+m x}{c}=1}..........(i)
We have the equation of the required pair of straight lines joining the origin to the intersection points of the  straight line \mathrm{ y=m x+c}  and the curve  \mathrm{ x^2+y^2=a^2}   as   \mathrm{ x^2+y^2-a^2\left(\frac{y+m x}{c}\right)^2=0}

i.e. \mathrm{ x^2\left(c^2-a^2 m^2\right)-2 m a^2 x y+y^2\left(c^2-a^2\right)=0}..............................(ii)
The above equation will represent two mutually perpendicular lines ifcoeff. of \mathrm{x}^2+  coeff. of  \mathrm{y}^2=0

\mathrm{\text { i.e. }\left(c^2-a^2 m^2\right)+\left(c^2-a^2\right)=0}   

\text { i.e. } 2 c^2=a^2\left(1+m^2\right) \quad

 which is the desired result. 

Posted by

Deependra Verma

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