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  • A line through meets the lines <img alt="\mathrm{x+3 y+2=0,2 x+y+4=0 \: and \: x-

A line through \mathrm{A(-5,-4)} meets the lines \mathrm{x+3 y+2=0,2 x+y+4=0 \: and \: x-y-5=0} at the points \mathrm{B, C\: and \: D} respectively. If
\mathrm{ \left(\frac{15}{A B}\right)^2+\left(\frac{10}{A C}\right)^2=\left(\frac{6}{A D}\right)^2 }find the equation of the line.

Option: 1

\mathrm{2 x-3 y+11=0}
 


Option: 2

\mathrm{2 x+3 y+22=0}
 


Option: 3

\mathrm{3 x-2 y+22=0}
 


Option: 4

none of these


Answers (1)

best_answer

Let the line through \mathrm{A(-5,-4)} makes an angle \mathrm{\theta\: with \: \mathrm{x}} - axis then the distance form its equation is
\mathrm{ \frac{x+5}{\cos \theta}=\frac{y+4}{\sin \theta} }        (1)

If \mathrm{ A B=r_1, A C=r_2, A D=r_3 } then for \mathrm{ \mathrm{B} }

For \mathrm{ C, \frac{x+5}{\cos \theta}=\frac{y+4}{\sin \theta}=r_2 \Rightarrow C \equiv\left(r_2 \cos \theta-5, r_2 \sin \theta-4\right) }

And for \mathrm{ D, \frac{x+5}{\cos \theta}=\frac{y+4}{\cos \theta}=r_3 \Rightarrow D \equiv\left(r_3 \cos \theta-5, r_3 \sin \theta-4\right) }

Given that \mathrm{ B, C, D } lie on lines

\mathrm{x+3 y+2=0,2 x+y+4=0 \: and \: x-y-5=0} respectively so that\mathrm{\left(r_1 \cos \theta-5\right)+3\left(r_1 \sin \theta-4\right)+2=0 \Rightarrow \frac{15}{r_1}=\cos \theta+3 \sin \theta }(2)
\mathrm{2\left(r_2 \cos \theta-5\right)+r_2(\sin \theta-4)+4=0 \Rightarrow \frac{10}{r_2}=2 \cos \theta+\sin \theta }(3) and \mathrm{\left(r_2 \cos \theta-5\right)-\left(r_3 \sin \theta-4\right)-5=0 \Rightarrow \frac{6}{r_3}=\cos \theta-\sin \theta }(4)

From (2), (3) and (4) \mathrm{ \left(\frac{15}{r_1}\right)^2+\left(\frac{10}{r_2}\right)^2=\left(\frac{6}{r_3}\right)^2 }

\mathrm{ \Rightarrow \quad(\cos \theta+3 \sin \theta)^2+(2 \cos \theta+\sin \theta)^2=(\cos \theta-\sin \theta)^2 }

\mathrm{ \text { or } 4 \cos ^2 \theta+9 \sin ^2 \theta+12 \sin \theta \cos \theta=0 }

\mathrm{ \Rightarrow \quad(2 \cos \theta+3 \sin \theta)^2=0 \Rightarrow \quad \frac{\cos \theta}{-3}=\frac{\sin \theta}{2}=k }(say)

\mathrm{ Putting \: \cos \theta=-3 k, \sin \theta=2 k in (1), the \: required \: equation\: is \: 2 x+3 y+22=0 }

Hence option 2 is correct.



 

Posted by

rishi.raj

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