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 circle has a center at \((-1, 2)\) and a radius of \(5\) units.
  • The distance from the center to the point \((2, -2)\) is calculated as \(\sqrt{(2 - (-1))^2 + (-2 - 2)^2} = \sqrt{3^2 + (-4)^2} = 5\).
  • Since the distance from the center equals the radius, the point lies on the circle.
  • Amit's calculation was numerically correct, but his final conclusion that the point does not lie on the circle is incorrect.

Explanation

  1. Verify the geometric properties and distance calculation.

The problem asks to verify if a point \(P = (2, -2)\) lies on a circle centered at \(C_0 = (-1, 2)\) with a diameter of \(10\) units.

First, we determine the properties of the circle:

  • The center of the circle is \((-1, 2)\).
  • The diameter \(d = 10\), so the radius \(r\) is \(\frac{d}{2} = \frac{10}{2} = 5\) units.

Next, we calculate the distance between the center \(C_0 = (-1, 2)\) and the point \(P = (2, -2)\) using the distance formula:
\[dist = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}\]
Substituting the given coordinates:
\[dist = \sqrt{(2 - (-1))^2 + (-2 - 2)^2} = \sqrt{(2 + 1)^2 + (-4)^2}\]
\[dist = \sqrt{3^2 + (-4)^2} = \sqrt{9 + 16} = \sqrt{25} = 5\]
The calculated distance is exactly \(5\), which is equal to the radius \(r = 5\). Since the distance from the center to the point is equal to the radius, the point \(P = (2, -2)\) lies on the circle.

  1. Evaluate Amit's work and conclusion.

Amit correctly identified the radius as \(5\). He also correctly calculated the distance from the center to the point as \(5\). However, Amit concluded that the point does not lie on the circle. Based on our calculation, the distance from the center to the point is indeed equal to the radius (\(5 = 5\)). Therefore, the point must lie on the circle, and Amit's conclusion is incorrect.

Reviewing the provided options to address his error:

  • "No, he should have used the origin as the center of the circle." (Incorrect)
  • "No, the radius is \(10\) units, not \(5\) units." (Incorrect)
  • "No, he did not calculate the distance correctly." (Incorrect, as the numerical calculation was correct.)
  • "Yes, the distance from the center to (2, -2) is not the same as the radius." (Incorrect, as the distance is exactly the same as the radius.)

Since all given options are factually incorrect in their assessment of the situation, we must conclude that Amit's work is flawed due to his final conclusion, despite the accurate numerical calculation. In the context of the provided choices, there appears to be no correct option. However, as the question demands an evaluation, we identify that the primary error lies in his conclusion, which contradicts his own calculation.

Examples

In real-life navigation or satellite tracking, verifying whether a target point lies on a circular path is common. For example, if a drone is programmed to maintain a circular flight path of radius \(500\text{ meters}\) centered at a control station, knowing the exact coordinates allows the navigation system to ensure the drone remains on course. Calculating the distance from the control station to the drone's current position confirms if the drone is correctly placed on its intended path. If the distance matches the radius, the drone is exactly where it needs to be!

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