10th Class Mathematics Coordinate Geometry Question Bank MCQs - Coordinate Geometry

If the point P (x, y) is equidistant from the points A(5, 1) and 5(1, 5), then

A) y=3x

B) x=y

C) x=-8y

D) -8x=y

Correct Answer: B

Solution :

Given, point \[P\left( x,\,y \right)\]is equidistant from the points A (5, 1) and B (1,5).

So, AP = BP

\[\Rightarrow \,\,\,A{{P}^{2}}=B{{P}^{2}}\][on squaring both sides]

\[\Rightarrow \,\,\,{{\left( x-5 \right)}^{2}}+{{\left( y-1 \right)}^{2}}={{\left( x-1 \right)}^{2}}+{{\left( y-5 \right)}^{2}}\]

[\[\because\] distance \[=\sqrt{{{\left( {{x}_{2}}-{{x}_{1}} \right)}^{2}}+{{\left( {{y}_{2}}-{{y}_{1}} \right)}^{2}}}\]]

\[\Rightarrow \,\,\,{{x}^{2}}+25-10x+{{y}^{2}}+1-2y\]

\[={{x}^{2}}+1-2x+{{y}^{2}}+25-10y\]

\[\left[ \because \,\,\,{{\left( a-b \right)}^{2}}={{a}^{2}}+{{b}^{2}}-2ab \right]\]

\[\Rightarrow \,\,\,\,\,-10x+2x=-10y+2y\]

\[\Rightarrow \,\,\,\,-8x=-8y\]

\[\Rightarrow \,\,\,\,x=y\]