Amit's work and his approach contain a mistake in the calculation of the distance between the center of the circle [tex]\((-1, 2)\)[/tex] and the given point [tex]\((2, -2)\)[/tex].
Let’s walk through the correct steps:
1. Understand the Problem: We need to find out whether the point [tex]\((2, -2)\)[/tex] is on the circle centered at [tex]\((-1, 2)\)[/tex] with a radius of 5 units.
2. Determine the Radius: The diameter of the circle is given as 10 units. Therefore, the radius of the circle is:
[tex]\[
\text{radius} = \frac{\text{diameter}}{2} = \frac{10}{2} = 5 \text{ units}
\][/tex]
3. Find the Distance From the Center to the Point: Use the distance formula to calculate the distance between the center [tex]\((-1, 2)\)[/tex] and the point [tex]\((2, -2)\)[/tex]:
[tex]\[
\text{Distance} = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}
\][/tex]
Substituting [tex]\((x_1, y_1) = (-1, 2)\)[/tex] and [tex]\((x_2, y_2) = (2, -2)\)[/tex]:
[tex]\[
\text{Distance} = \sqrt{(2 - (-1))^2 + (-2 - 2)^2}
\][/tex]
Simplifying inside the square root:
[tex]\[
\text{Distance} = \sqrt{(2 + 1)^2 + (-2 - 2)^2} = \sqrt{3^2 + (-4)^2}
\][/tex]
Calculating further, we get:
[tex]\[
\text{Distance} = \sqrt{9 + 16} = \sqrt{25} = 5 \text{ units}
\][/tex]
4. Compare Distance with Radius: The calculated distance from the center [tex]\((-1, 2)\)[/tex] to the point [tex]\((2, -2)\)[/tex] is 5 units.
Since the calculated distance (5 units) is equal to the radius of the circle (5 units), the point [tex]\((2, -2)\)[/tex] lies on the circle.
Therefore, Amit’s conclusion is incorrect. The correct conclusion should be:
[tex]\[
\text{Yes, the point } (2, -2) \text{ does lie on the circle because the distance from the center is equal to the radius.}
\][/tex]
Amit did not calculate the distance correctly. The correct distance calculation shows that the point [tex]\((2, -2)\)[/tex] is indeed on the circle, as the distance is exactly 5 units, which matches the radius of the circle.
