If sum of the perpendicular distances of a variable point `P (x , y)`from the lines `x + y 5 = 0`and `3x 2y +7 = 0`is always 10. Show that P must move on a line.

Explanation

Given, lines `x + y 5 = 0`and `3x +2y +7 = 0`

let `x + y 5 = 0`...(1)

`3x 2y +7 = 0`....(2)

The perpendicular distance of P(x,y) from lines (1) and (2)
are respectively given by

`d_1​=∣x+y−5∣/sqrt(1^2+1^2)​`

and `d_2​=∣3x−2y+7∣​/sqrt(3^2+2^2)`

it is given that `d_1​+d_2​=10`

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