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 :

  • Verify that the radius is \(5\) and the distance from the center \((-1, 2)\) to \((2, -2)\) is also \(5\).
  • Recognize that since the distance equals the radius, the point \((2, -2)\) must lie on the circle.
  • Identify that Amit's work is incorrect because his final conclusion contradicts his accurate mathematical calculation.
  • Conclude that the point is indeed on the circle, making his assertion false.

Explanation

  1. Analyze the radius and distance calculations.

To determine if the point \(P = (2, -2)\) lies on the circle centered at \(C = (-1, 2)\) with a diameter of \(10\) units, we first need to verify the radius of the circle and the distance from the center to the point \(P\).

The radius \(r\) is half of the diameter:
\[r = \frac{10}{2} = 5\]
This part of Amit's work is correct.

Next, we calculate the distance between the center \(C = (-1, 2)\) and the point \(P = (2, -2)\) using the distance formula \(d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}\):
\[d = \sqrt{(2 - (-1))^2 + (-2 - 2)^2}\]
\[d = \sqrt{(3)^2 + (-4)^2}\]
\[d = \sqrt{9 + 16} = \sqrt{25} = 5\]
Amit's calculation of the distance is also correct.

  1. Evaluate Amit's conclusion and overall correctness.

A point lies on a circle if and only if the distance from the center of the circle to that point is equal to the radius of the circle.

In this case, we have:

  • The distance from the center \(C\) to the point \(P\) is \(d = 5\).
  • The radius of the circle is \(r = 5\).

Since \(d = r = 5\), the point \(P = (2, -2)\) must lie on the circle.

Amit concludes that the point \((2, -2)\) does not lie on the circle. This conclusion contradicts his own findings, as he correctly calculated both the radius and the distance to be \(5\). Therefore, Amit's work is incorrect because his final conclusion is logically flawed despite his accurate calculations. None of the provided options accurately describe why his work is wrong, as the error lies in his final interpretation, not in his radius or distance calculation.

  1. Final Conclusion.

Amit's work is incorrect because, although he correctly identified the radius and calculated the distance between the center and the point, he reached the wrong conclusion. Since the distance from the center to the point is equal to the radius, the point \((2, -2)\) is on the circle. Therefore, he incorrectly stated that the point does not lie on the circle.

The final answer is that Amit's work is not correct.

Examples

In real-world applications like GPS navigation or robotics, verifying if a point is within a specific range is crucial. For example, if a drone is programmed to maintain a circular perimeter around a central beacon, we use the distance formula to check if the drone's coordinates satisfy the equation \((x - h)^2 + (y - k)^2 = r^2\). If the distance from the beacon \((h, k)\) to the drone \((x, y)\) equals the radius \(r\), we confirm it is exactly on the boundary of its permitted zone. Understanding this logic prevents errors in automated navigation systems.

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