Name: The Line Family
Uploaded: 2019-07-19
Duration: 4 min 33 s
Description: A family of lines is a set of lines that have something in common with each other. Learn more about the family of lines by watching this video.

Question

Given $n$ straight lines and a fixed point O. A straight line meeting these lines is drawn through in the points $R_{1}, R_{2}, R_{3}, . . . R_{n,}$ and on it is taken a point $R$ such that $\frac{n}{O R} = \frac{1}{O R_{1}} + \frac{1}{O R_{2}} + \frac{1}{O R_{3}} + . . . . . . . + \frac{1}{O R_{n}};$ show that locus of $R$ is a straight line.

Accepted Answer

Let the equations of the straight lines are $p_{i} x + q_{i} y + c_{i} = 0$ ,

where $i = 1, 2, 3, . . . . . . . n .$

Their polar equation will be $\frac{1}{r_{i}} = \frac{p_{i} \cos α + q_{i} \sin α}{c_{i}}$ .

If the line makes an angle $α$ with initial line x-axis then on the basis of hypothesis, we have

$\frac{n}{r} = \sum_{i = 1}^{n} \frac{1}{r_{i}}$

$\frac{n}{r} = \frac{p_{1} \cos α + q_{1} \sin α}{c_{1}} + \frac{p_{2} \cos α + q_{2} \sin α}{c_{2}} + . . . . . . . . .$

$n = (r \cos α) (\frac{p_{1}}{c_{1}} + \frac{p_{2}}{c_{2}} + . . . . .) + (r \sin α) (\frac{q_{1}}{c_{1}} + \frac{q_{2}}{c_{2}} + . . . . .) + . . . . . . . . . . . . .$

Let $x = r \cos α, y = r \sin α$

$n = x (\frac{p_{1}}{c_{1}} + \frac{p_{2}}{c_{2}} + . . . . .) + y (\frac{q_{1}}{c_{1}} + \frac{q_{2}}{c_{2}} + . . . . .) + . . . . . . . . .$

$x (\sum \frac{p_{1}}{c_{1}}) + y (\sum \frac{q_{1}}{c_{1}}) - n = 0$

So this is the format of the straight line $a x + b y + c = 0$ .

Hence, proved.

Important Questions on The Straight Line (Continued)

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Given n straight lines and a fixed point O. A straight line meeting these lines is drawn through in the points R1, R2, R3, ...Rn, and on it is taken a point R such that nOR=1OR1+1OR2+1OR3+.......+1ORn; show that locus of R is a straight line.

Important Questions on The Straight Line (Continued)

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