Question

does the point (10,3) lie on the circle that passes through the point (2,9) with center (3,2)? explain. no, because the length of the radius of the circle is ( 5sqrt{2} ) and ( sqrt{(-10-(-3))^{2}+(3 - 2)^{2}}
eq5sqrt{2} ). no, because the length of the radius of the circle is ( sqrt{74} ) and ( sqrt{(-10-(-3))^{2}+(3 - 2)^{2}}
eqsqrt{74} ). yes, because the length of the radius of the circle is ( sqrt{170} ) and ( sqrt{(-10 - 3)^{2}+(3 - 2)^{2}}=sqrt{170} ). yes, because the length of the radius of the circle is ( 5sqrt{2} ) and ( sqrt{(-10-(-3))^{2}+(3 - 2)^{2}}=5sqrt{2} ).

Explanation:

Step1: Calculate the circle radius

The radius \( r \) is the distance from center \((3,2)\) to point \((2,9)\):

$$ r = \sqrt{(2-3)^2 + (9-2)^2} = \sqrt{(-1)^2 + 7^2} = \sqrt{1 + 49} = \sqrt{50} = 5\sqrt{2} $$

Step2: Check distance from center to (10,3)

Distance \( d \) from \((3,2)\) to \((10,3)\):

$$ d = \sqrt{(10-3)^2 + (3-2)^2} = \sqrt{7^2 + 1^2} = \sqrt{49 + 1} = \sqrt{50} = 5\sqrt{2} $$

Step3: Compare distance and radius

Since \( d = r = 5\sqrt{2} \), the point lies on the circle. (Note: The option may have a typo in coordinates, but the radius and equality match the fourth option's logic.)

Answer:

Yes, because the length of the radius of the circle is \(5\sqrt{2}\) and \(\sqrt{(-10 - (-3))^2 + (3 - 2)^2} = 5\sqrt{2}\). (Correcting the coordinate typo in the option, the core equality holds.)