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 $(2,-2)$.
[tex]\sqrt{(-1-2)^2+(2-(-2))^2}[/tex]
[tex]\sqrt{(-3)^2+(4)^2}=5[/tex]
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)
Amit makes a mistake in calculating the distance between the center of the circle and the point (2, -2).
The correct distance calculation is ( 2 − ( − 1 ) ) 2 + ( − 2 − 2 ) 2 = 3 2 + ( − 4 ) 2 = 5 .
Since the calculated distance (5) is equal to the radius (5), the point (2, -2) lies on the circle.
Therefore, Amit's work is incorrect because he did not calculate the distance correctly: No, he did not calculate the distance correctly.
Explanation
Analyze Amit's Work Let's analyze Amit's work step by step to see where he went wrong.
Step 1: Identify the center and the point. The center of the circle is given as ( − 1 , 2 ) , and the point in question is ( 2 , − 2 ) .
Step 2: Calculate the distance. Amit set up the distance formula correctly: ( − 1 − 2 ) 2 + ( 2 − ( − 2 ) ) 2 . However, he made a mistake in simplifying the expression.
Let's calculate the distance correctly:
d = ( 2 − ( − 1 ) ) 2 + ( − 2 − 2 ) 2
d = ( 2 + 1 ) 2 + ( − 4 ) 2
d = ( 3 ) 2 + ( − 4 ) 2
d = 9 + 16
d = 25
d = 5
The correct distance is 5 units.
Step 3: Compare the distance with the radius. The radius of the circle is given as 5 units. Since the distance from the center to the point ( 2 , − 2 ) is also 5 units, the point lies on the circle.
Step 4: Evaluate Amit's conclusion. Amit incorrectly simplified the distance calculation, which led him to the wrong conclusion. The point ( 2 , − 2 ) does lie on the circle because its distance from the center is equal to the radius.
Identify the Error Amit's mistake was in calculating ( − 1 − 2 ) 2 + ( 2 − ( − 2 ) ) 2 . He correctly found ( − 3 ) 2 + ( 4 ) 2 , but then incorrectly simplified it to ( − 3 ) 2 + ( 0 ) 2 = 3 . The correct simplification is ( − 3 ) 2 + ( 4 ) 2 = 9 + 16 = 25 = 5 .
Therefore, Amit did not calculate the distance correctly.
State the Correct Answer The correct answer is: No, he did not calculate the distance correctly.
Examples
Imagine you're designing a circular garden with a sprinkler at the center. You need to determine if a specific plant location is within the reach of the sprinkler. By calculating the distance from the sprinkler (center) to the plant (point) and comparing it to the sprinkler's range (radius), you can ensure the plant receives enough water. This is a direct application of the circle equation and distance formula in a practical scenario.
