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Question

$$(6,0),(0,6) $$ and $$(7,7)$$ are the vertices of a triangle. The circle inscribed in the triangle has equation.


A
x2+y29x+9y+36=0
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B
x2+y29x9y+36=0
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C
x2+y2+9x9y+36=0
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D
x2+y29x9y36=0
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Solution

The correct option is B $$x^2+y^2-9x-9y+36=0$$
Let, $$\triangle ABC$$ has the vertices $$A(6,0), B(0,6), C(7,7)$$ and $$AB, BC, CA$$ are the sides with distances $$c, a, b$$ respectively.

$$\therefore$$ Using distance formula, $$a=\left| BC \right| =\sqrt { { (7-0) }^{ 2 }+{ (7-6) }^{ 2 } } =5\sqrt { 2 } \\ b=\left| CA \right| =\sqrt { { (7-6) }^{ 2 }+{ (7-0) }^{ 2 } } =5\sqrt { 2 } \\ c=\left| AB \right| =\sqrt { { (6-0) }^{ 2 }+{ (0-6) }^{ 2 } } =6\sqrt { 2 } $$

Now, assuming the In-centre of the circle as $$O$$ we get, 
                         $$O=\left( \dfrac { a{ x }_{ 1 }+b{ x }_{ 2 }+c{ x }_{ 3 } }{ a+b+c } ,\dfrac { a{ y }_{ 1 }+b{ y }_{ 2 }+c{ y }_{ 3 } }{ a+b+c }  \right) \\$$

$$ =\left( \dfrac { 5\sqrt { 2 } (6)+5\sqrt { 2 } (0)+6\sqrt { 2 } (7) }{ 5\sqrt { 2 } +5\sqrt { 2 } +6\sqrt { 2 }  } ,\dfrac { 5\sqrt { 2 } (0)+5\sqrt { 2 } (6)+6\sqrt { 2 } (7) }{ 5\sqrt { 2 } +5\sqrt { 2 } +6\sqrt { 2 }  }  \right) \\$$

$$ \Rightarrow O=\left( \dfrac { 9 }{ 2 } ,\dfrac { 9 }{ 2 }  \right) $$
 and,
             
The inradius r is equal to the perpendicular distance from incenter to any of the sides

Equation of the line $$AB$$

$$y-0=\dfrac{6-0}{0-6}(x-6)$$   $$[two-point\ form]$$

$$y=\dfrac{6}{-6}(x-6)$$

$$y=-(x-6)$$

$$x+y-6=0$$

So the inradius is

$$r=\dfrac{|\dfrac{9}{2}+\dfrac{9}{2}-6|}{\sqrt{1+1}}$$
-
$$r=\dfrac{|9-6|}{\sqrt{2}}$$

$$r=\dfrac{|3|}{\sqrt{2}}$$

$$\therefore\ r=\dfrac{|3|}{\sqrt{2}}$$

The equation of the incircle is

$$(x-\dfrac{9}{2})^2+(y-\dfrac{9}{2})^2=\left(\dfrac{|3|}{\sqrt{2}} \right)^2$$

$$x^2+\dfrac{81}{4}-9x+y^2+\dfrac{81}{4}-9y=\dfrac{9}{2}$$

$$x^2+-9x+y^2-9y+\dfrac{81}{4}+\dfrac{81}{4}-\dfrac{9}{2}=0$$

$$x^2+y^2-9x-9y+\dfrac{81}{2}-\dfrac{9}{2}=0$$

$$x^2+y^2-9x-9y+\dfrac{81-9}{2}=0$$

$$x^2+y^2-9x-9y+\dfrac{72}{2}=0$$

$$x^2+y^2-9x-9y+36=0$$


1129415_1105286_ans_39e60eb2ae9f43f1a112c9b7e4f7b73d.png

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