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(B) The circle passes through the points $A(3,4)$,$B(3,2)$,and $C(1,4)$.Since $AB$ is a vertical line segment $(x=3)$ and $AC$ is a horizontal line se

Vedclass · Accepted Answer

$18$

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(B) The circle passes through the points $A(3,4)$,$B(3,2)$,and $C(1,4)$.Since $AB$ is a vertical line segment $(x=3)$ and $AC$ is a horizontal line segment $(y=4)$,the angle at $A(3,4)$ is $90^\circ$.Thus,$BC$ is the diameter of the circle.The midpoint of $BC$ is the center $(h, k) = (\frac{3+1}{2}, \frac{2+4}{2}) = (2, 3)$.The radius $r$ is the distance from the center $(2, 3)$ to $(3, 4)$:$r = \sqrt{(3-2)^2 + (4-3)^2} = \sqrt{1^2 + 1^2} = \sqrt{2}$.The parametric equations are $x = 2 + \sqrt{2} \cos 	heta$ and $y = 3 + \sqrt{2} \sin 	heta$.Comparing with $x = a + r \cos 	heta$ and $y = b + r \sin 	heta$,we get $a = 2$,$b = 3$,and $r = \sqrt{2}$.Then,$b^a \cdot r^a = 3^2 \cdot (\sqrt{2})^2 = 9 \cdot 2 = 18$.

If the parametric equations of the circle passing through the points $(3,4)$,$(3,2)$,and $(1,4)$ are $x=a+r \cos \theta$ and $y=b+r \sin \theta$,then find the value of $b^{a} r^{a}$.

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