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  • Find the locus of the middle points of the segment of a line passing through the point of intersection of the lines <img alt="\mathrm{ax + by + c = 0}" src="https://entrancecorner.oncodecogs.com/gi

Find the locus of the middle points of the segment of a line passing through the point of intersection of the lines \mathrm{ax + by + c = 0} and \mathrm{lx + my + n = 0}, which is intercepted between the axes.

Option: 1

\mathrm{lcx+ nby}


Option: 2

\mathrm{\left ( lc-an \right )x+\left ( nb-mc \right )y}


Option: 3

\mathrm{\left ( lc+an \right )x+\left ( nb+mc \right )y}


Option: 4

None of these 


Answers (1)

best_answer

Any line (say / = 0) passing through the point of intersection of \mathrm{ax + by + c = 0} and \mathrm{lx + my + n = 0 \; is\; (ax + by + c) + \lambda (lx + my + n) = 0,} where \lambda is any real number. Point of intersection of \mathrm{I=0} with axes are \mathrm{\left(-\frac{c+\lambda n}{a+\lambda l}, 0\right) \text { and }\left(0,-\frac{c+\lambda n}{b+\lambda m}\right)}

Let the mid point be (h, k). Then \mathrm{\mathrm{h}=-\frac{1}{2} \frac{\mathrm{c}+\lambda \mathrm{n}}{\mathrm{a}+\lambda l} \text { and } \mathrm{k}=-\frac{1}{2} \frac{\mathrm{c}+\lambda \mathrm{n}}{\mathrm{b}+\lambda \mathrm{m}}}

Eleminating \lambda , we get

\mathrm{\frac{2 \mathrm{ah}+\mathrm{c}}{2 \mathrm{~h} l+\mathrm{n}}=\frac{2 \mathrm{~kb}+\mathrm{c}}{2 \mathrm{~km}+\mathrm{n}}}

The required locus is: \mathrm{2(a m-l b) x y=(l c-a n) x+(n b-m c) y}

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manish

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