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Question

If sum of the perpendicular distances of a variable point P(x,y) from the lines x+y5 = 0 and 3x2y+7 = 0 is always 10. Show that P must move on a line.

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Solution

The perpendicular distance of P(x,y) from lines (1) and (2) are respectively given by
d1=|x+y5|(1)2+(1)2andd2=|3x2y+7|(3)2+(2)2
i.e.,d1=|x+y5|2andd2=|3x2y+7|13
it is given that d1+d2=10
|x+y5|2+|3x2y+7|13=10
13|x+y5|+2|3x2y+7|1026=0
[Assuming (x+y-5) and (3x-2y+7) are positive]
13x+13y513+32x22y+721026=0
x(13+32)+y(1322)+(725131026)=0,
which is the equation of a line similarly, we can obtain the equation of line for any signs of (x+y5) and (3x2y+7)
Thus, The point P must move on a line.

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