Answered

A circle centered at (-1, 2) has a diameter of 10 units. Amit wants to determine whether (2, -2) is also on the circle. His work is shown below.



The radius is 5 units.

Find the distance from the center to (2, -2).

$\begin{array}{l}

\sqrt{(-1-2)^2+(2-(-2))^2} \\

\sqrt{(-3)^2+(0)^2}=3

\end{array}$



The point (2, -2) doesn't lie on the circle because the calculated distance should be the same as the radius.



Is Amit's work correct?

ANo, he should have used the origin as the center of the circle.
BNo, the radius is 10 units, not 5 units.
CNo, he did not calculate the distance correctly.
DYes, the distance from the center to (2, -2) is not the same as the radius.

Verified Answer :

  • The radius of the circle is calculated as half the diameter, which is \(\frac{10}{2} = 5\).
  • The distance from the center \((-1, 2)\) to the point \((2, -2)\) is calculated using the distance formula, resulting in \(\sqrt{(2 - (-1))^2 + (-2 - 2)^2} = 5\).
  • Since the distance equals the radius, the point lies on the circle, making Amit's work correct.
  • Amit's final conclusion: \(\boxed{\text{Amit's work is correct}}\).

Explanation

  1. Verify Amit's calculation and reasoning.

The problem asks to verify if the point \((2, -2)\) lies on a circle centered at \((-1, 2)\) with a diameter of \(10\) units.

First, we determine the radius \(r\) of the circle using the given diameter \(D = 10\):
\[r = \frac{D}{2} = \frac{10}{2} = 5\]
The radius of the circle is \(5\) units.

Next, we calculate the distance \(d\) between the center \(C = (-1, 2)\) and the point \(P = (2, -2)\) using the distance formula:
\[d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}\]
Substituting the given coordinates:
\[d = \sqrt{(2 - (-1))^2 + (-2 - 2)^2} = \sqrt{(3)^2 + (-4)^2} = \sqrt{9 + 16} = \sqrt{25} = 5\]
The distance from the center to the point \((2, -2)\) is \(5\) units.

Since the distance \(d = 5\) is equal to the radius \(r = 5\), the point \((2, -2)\) satisfies the condition for lying on the circle. Therefore, Amit's calculations are correct. Looking at the provided options, they all contain errors in their logical statement or claim that Amit's work is incorrect. Given that Amit's work is indeed correct, it appears the provided options do not include a correct positive evaluation. However, based on the steps taken, Amit's logic is perfectly sound.

Examples

Understanding the relationship between a point and a circle is essential in fields like navigation and computer graphics. For example, if a ship is moving on a radar screen centered at a specific location, engineers use the distance formula to determine if the ship has reached a certain range, which is defined by a circular boundary. If the distance from the center to the ship's coordinates equals the radius, the ship is exactly on the boundary. This allows for precise monitoring and tracking of objects within a defined zone.

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