A line is such that its segment between the lines `5x-y+4=0` and `3x+4y-4=0` is bisected at the point (1,5). Obtain its equation.

Explanation

Let PQ be the intercept between the lines `5x−y−4=0 ` and `3x+4y−4=0 ` passing through point A`(1,5)` and making angle `theta `with positive x-axis.

So, its equation is,

`frac{x-1}{costheta}=frac{y-5}{sin theta}`

`implies frac{y-5}{x-1}=tan theta`

Let AP=AQ=`r`

So, coordinates of P and Q are `(1pm r cos theta ,5pm r sin theta)`respectively.

Clearly, P and Q lie on `5x−y−4=0 ` and `3x+4y−4=0 ` respectively.

`therefore 5(1+r cos theta)-5-r sin theta-4=0` and `3(1-r cos theta)+4(5-r sin theta)-4=0`

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