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If the points A(4, 3) and B(x, 5) are on a circle with center C(2, 3), find the values of x.

Coordinate Geometry

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Answer

Given,

A(4, 3) and B(x, 5) are on a circle with center C(2, 3).

If the points A(4, 3) and B(x, 5) are on a circle with center C(2, 3), find the values of x. Coordinate Geometry, ML Aggarwal Understanding Mathematics Solutions ICSE Class 9.

∴ AC = BC [Radius of same circle]

By distance formula,

d = (x2−x1)2+(y2−y1)2\sqrt{(x2 - x1)^2 + (y2 - y1)^2}

Substituting values in above equation, we get :

⇒(2−4)2+(3−3)2=(2−x)2+(3−5)2⇒(−2)2=4+x2−4x+(−2)2⇒4=4+x2−4x+4\Rightarrow \sqrt{(2 - 4)^2 + (3 - 3)^2} = \sqrt{(2 - x)^2 + (3 - 5)^2} \\[1em] \Rightarrow \sqrt{(-2)^2} = \sqrt{4 + x^2 - 4x + (-2)^2} \\[1em] \Rightarrow \sqrt{4} = \sqrt{4 + x^2 - 4x + 4}

On squaring both sides,

⇒4=x2−4x+8⇒x2−4x+4=0⇒x2−2x−2x+4=0⇒x(x−2)−2(x−2)=0⇒(x−2)(x−2)=0⇒x−2=0⇒x=2.\Rightarrow 4 = x^2 - 4x + 8 \\[1em] \Rightarrow x^2 - 4x + 4 = 0 \\[1em] \Rightarrow x^2 - 2x - 2x + 4 = 0 \\[1em] \Rightarrow x(x - 2) - 2(x - 2) = 0 \\[1em] \Rightarrow (x - 2)(x - 2) = 0 \\[1em] \Rightarrow x - 2 = 0 \\[1em] \Rightarrow x = 2.

Hence, value of x = 2.

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