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Question: If it is possible to draw a triangle which circumscribe the circle\({(x - (\alpha - 2\beta ))^2} + {...

If it is possible to draw a triangle which circumscribe the circle(x−(α−2β))2+(y−(α+β))2=1{(x - (\alpha - 2\beta ))^2} + {(y - (\alpha + \beta ))^2} = 1 and is inscribed byx2+y2−2x−4y+1=0{x^2} + {y^2} - 2x - 4y + 1 = 0, then
1. β=−13\beta = \dfrac{{ - 1}}{3}
2. β=23\beta = \dfrac{2}{3}
3. α=53\alpha = \dfrac{5}{3}
4. α=−52\alpha = \dfrac{{ - 5}}{2}

Explanation

Solution

Hint: Focus on showing that the circumcenter and incenter are at the same point by using the formula of distance between incenter and circumcenter i.e. Distance = R2−2rR\sqrt {{R^2} - 2rR} and show that the distance between incenter and circumcenter is 0 and find the value of α\alpha andβ\beta ,

Complete step-by-step answer:

Let’s construct the rough diagram using the given information

Equation of incircle is (x−(α−2β))2+(y−(α+β))2=1{(x - (\alpha - 2\beta ))^2} + {(y - (\alpha + \beta ))^2} = 1

And equation of circumcircle is x2+y2−2x−4y+1=0{x^2} + {y^2} - 2x - 4y + 1 = 0

⇒ \Rightarrow (x−1)2+(y−2)2=4{(x - 1)^2} + {(y - 2)^2} = 4

So the radius of incircle r = 1 and the radius of circumcircle R = 2

Since we have the radius of incircle and circumcircle

Applying the formula of distance between P and O i.e. distance = R2−2rR\sqrt {{R^2} - 2rR}

Distance = 22−2×1×2\sqrt {{2^2} - 2 \times 1 \times 2}

Distance = 0

Since the distance between P and O is 0

Therefore circumcenter and incenter are at the same point

The general equation of the circle is (x−h)2+(y−k)2=r2{(x - h)^2} + {(y - k)^2} = {r^2}where (h, k) is the center of the circle.

Since both incircle and circumcircle have the same center therefore

By the of equation 1 and 2

α−2β=1\alpha - 2\beta = 1 -(Equation 1)

α+β=2\alpha + \beta = 2 -(Equation 2)

Subtracting equation 2 by equation 1

α+β−α+2β=2−1\alpha + \beta - \alpha + 2\beta = 2 - 1

⇒\Rightarrow β=13\beta = \dfrac{1}{3}

Putting the value of α\alpha in equation 1

α−2×13=1\alpha - 2 \times \dfrac{1}{3} = 1

Therefore value α=53\alpha = \dfrac{5}{3}.

Note: In these types of questions constructing a rough diagram using the given information in the question using the distance formula to find the distance between P and O. Then show that both incircle and circumcircle have the same center next comparing the equations of both circles and find the value of α\alpha andβ\beta .