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 $(2,-2)$.
$\sqrt{(-1-2)^2+(2-(-2))^2}$
$\sqrt{(-3)^2+(4)^2} = 5$
The point $(2,-2)$ lies on the circle because the calculated distance is the same as the radius.
Is Amit's work correct?
A. No, he should have used the origin as the center of the circle.
B. No, the radius is 10 units, not 5 units.
C. No, he did not calculate the distance correctly.
D. Yes, the distance from the center to $(2,-2)$ is the same as the radius.
Answer (1)
Determine the radius of the circle using the given diameter: r = 2 10 = 5 .
Calculate the distance between the center ( − 1 , 2 ) and the point ( 2 , − 2 ) using the distance formula: d = ( 2 − ( − 1 ) ) 2 + ( − 2 − 2 ) 2 = 5 .
Compare the calculated distance with the radius: since d = r = 5 , the point lies on the circle.
Conclude that Amit's work is incorrect because he miscalculated the distance. The correct answer is: No, he did not calculate the distance correctly. N o , h e d i d n o t c a l c u l a t e t h e d i s t an cecorrec tl y .
Explanation
Analyze the problem The problem asks us to verify Amit's work in determining whether the point ( 2 , − 2 ) lies on a circle centered at ( − 1 , 2 ) with a diameter of 10 units. Amit calculated the distance between the center and the point and compared it to the radius. We need to check if his calculations and conclusion are correct.
Find the radius First, let's find the radius of the circle. The diameter is given as 10 units, so the radius r is half of the diameter: r = 2 10 = 5 The radius is 5 units.
Calculate the distance Next, we need to calculate the distance d between the center of the circle ( − 1 , 2 ) and the point ( 2 , − 2 ) using the distance formula: d = ( x 2 − x 1 ) 2 + ( y 2 − y 1 ) 2 Substitute the coordinates: d = ( 2 − ( − 1 ) ) 2 + ( − 2 − 2 ) 2 Simplify the expression: d = ( 3 ) 2 + ( − 4 ) 2 = 9 + 16 = 25 = 5 The distance between the center and the point is 5 units.
Analyze Amit's calculation Now, let's analyze Amit's calculation. Amit wrote: ( − 1 − 2 ) 2 + ( 2 − ( − 2 ) 2 which simplifies to ( − 3 ) 2 + ( 0 ) 2 = 3 .
Amit incorrectly calculated the y-component of the distance. He made a mistake by squaring only the inner -2 instead of calculating 2 - (-2) = 4 first, and then squaring it. The correct calculation should be: d = ( − 1 − 2 ) 2 + ( 2 − ( − 2 ) ) 2 = ( − 3 ) 2 + ( 4 ) 2 = 9 + 16 = 25 = 5 Amit's calculation is incorrect.
Conclusion Since the distance between the center of the circle and the point ( 2 , − 2 ) is 5 units, which is equal to the radius of the circle, the point ( 2 , − 2 ) lies on the circle. Amit's conclusion that the point doesn't lie on the circle is incorrect because his distance calculation was wrong.
Final Answer Amit's work is incorrect because he did not calculate the distance correctly. The correct distance is 5, which is equal to the radius, meaning the point (2, -2) lies on the circle.
Examples
Understanding circles and distances is crucial in many real-world applications. For example, consider a GPS system that uses satellites to determine your location. The GPS calculates the distance between your device and several satellites. Knowing the distances and the satellites' positions, the GPS can pinpoint your location by finding the intersection of these distances, which can be visualized as circles (or spheres in 3D). This principle is also used in various navigation systems and surveying techniques.
