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Analytical Geometry> If the sum of distances of a point from t...

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If the sum of distances of a point from two perp[endicular lines in a plane is 1, the find the locus of the point.

vardhan , 10 Years ago

Grade 12

1 Answers

Jitender Singh

Last Activity: 10 Years ago

Ans: Line

Sol:

Let the coordinates of the point be (h, k). The equation of two perpendicular lines:

$ax+by+c=0$ ….......(1)

$-bx+ay+c^{'}=0$ …...........(2)

Whera a, b, c & c^’are constants. Let distance of point form (1) & (2) be d₁& d_2.

$d_{1}+d_{2}=1$

$d_{1}=\frac{|ah+bk+c|}{\sqrt{a^{2}+b^{2}}}$

$d_{2}=\frac{|-bh+ak+c^{'}|}{\sqrt{a^{2}+b^{2}}}$

$\frac{|ah+bk+c|}{\sqrt{a^{2}+b^{2}}}+\frac{|-bh+ak+c^{'}|}{\sqrt{a^{2}+b^{2}}} = 1$

$|ah+bk+c|+|-bh+ak+c^{'}| = \sqrt{a^{2}+b^{2}}$

$(a-b)h+(a+b)k+c+c^{'}=\sqrt{a^{2}+b^{2}}$

$h\rightarrow x, b\rightarrow y$

$(a-b)x+(a+b)y+c+c^{'}=\sqrt{a^{2}+b^{2}}$

There would be more solutions but all will be straight line.

Thanks & Regards

Jitender Singh

IIT Delhi

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