Question #294741

𝐴(1,βˆ’2) is a point on the circle (π‘₯βˆ’3)2+ (𝑦+1)2= 5

a. State the coordinates of the centre of the circle and hence find the coordinates of the point 𝐡 where 𝐴𝐡 is the diameter of the circle.

b. 𝐢(2,1) also lies on the circle. Use coordinate geometry to verify that angle 𝐴𝐢𝐡 = 900


1
Expert's answer
2022-02-07T17:49:19-0500

a.


(π‘₯βˆ’3)2+(𝑦+1)2=5(π‘₯βˆ’3)^2+ (𝑦+1)^2= 5

The coordinates of the centre of the circle are D(3,βˆ’1).D(3, -1).

If ABAB is the diameter of the circle, then DD is the midpoint of ABAB


xD=xA+xB2=>xB=2xDβˆ’xAx_D=\dfrac{x_A+x_B}{2}=>x_B=2x_D-x_A

yD=yA+yB2=>yB=2yDβˆ’yAy_D=\dfrac{y_A+y_B}{2}=>y_B=2y_D-y_A

xB=2(3)βˆ’1=5,yB=2(βˆ’1)βˆ’(βˆ’2)=0x_B=2(3)-1=5, y_B=2(-1)-(-2)=0

Point B(5,0)B(5, 0)


b.

Point C(2,1)C(2, 1)


CAβ†’=⟨1βˆ’2,βˆ’2βˆ’1⟩\overrightarrow{CA}=\langle1-2, -2-1\rangle

CBβ†’=⟨5βˆ’2,0βˆ’1⟩\overrightarrow{CB}=\langle5-2, 0-1\rangle

CAβ†’β‹…CBβ†’=βˆ’1(3)+(βˆ’3)(βˆ’1)=βˆ’3+3=0\overrightarrow{CA}\cdot \overrightarrow{CB}=-1(3)+(-3)(-1)=-3+3=0

∣CAβ†’βˆ£=ΜΈ0,∣CBβ†’βˆ£=ΜΈ0|\overrightarrow{CA}|\not=0, |\overrightarrow{CB}|\not=0

Therefore CAβ†’βŠ₯CBβ†’,\overrightarrow{CA}\perp \overrightarrow{CB}, and ∠ACB=90Β°.\angle ACB=90\degree.



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