To resolve whether Amit's work is correct or not, let's go through the correct process to determine if the point [tex]\((2, -2)\)[/tex] lies on the circle centered at [tex]\((-1, 2)\)[/tex] with a diameter of 10 units.
1. Calculate the radius of the circle:
Given the diameter of the circle is 10 units, we can find the radius [tex]\( r \)[/tex] by dividing the diameter by 2:
[tex]\[
r = \frac{10}{2} = 5 \text{ units}
\][/tex]
2. Calculate the distance from the center of the circle [tex]\((-1, 2)\)[/tex] to the point [tex]\((2, -2)\)[/tex]:
The distance formula between two points [tex]\((x_1, y_1)\)[/tex] and [tex]\((x_2, y_2)\)[/tex] is given by:
[tex]\[
\text{Distance} = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}
\][/tex]
Plugging in the coordinates of the center [tex]\((-1, 2)\)[/tex] and the point [tex]\((2, -2)\)[/tex]:
[tex]\[
\text{Distance} = \sqrt{(2 - (-1))^2 + (-2 - 2)^2}
\][/tex]
Simplifying the terms inside the square root:
[tex]\[
\text{Distance} = \sqrt{(2 + 1)^2 + (-2 - 2)^2}
\][/tex]
[tex]\[
\text{Distance} = \sqrt{3^2 + (-4)^2}
\][/tex]
[tex]\[
\text{Distance} = \sqrt{9 + 16}
\][/tex]
[tex]\[
\text{Distance} = \sqrt{25}
\][/tex]
[tex]\[
\text{Distance} = 5 \text{ units}
\][/tex]
3. Compare the calculated distance with the radius:
The radius of the circle is 5 units, and the calculated distance from the center [tex]\((-1, 2)\)[/tex] to the point [tex]\((2, -2)\)[/tex] is also 5 units. Since the calculated distance matches the radius, the point [tex]\((2, -2)\)[/tex] lies on the circle.
4. Review Amit's work:
Amit's work hints that he calculated the distance as:
[tex]\[
\sqrt{(-1 - 2)^2 + (2 - (-2))^2}
\][/tex]
This simplifies to:
[tex]\[
\sqrt{(-3)^2 + (4)^2}
\][/tex]
[tex]\[
\sqrt{9 + 16}
\][/tex]
[tex]\[
\sqrt{25}
\][/tex]
[tex]\[
5 \text{ units}
\][/tex]
As shown, Amit did calculate the distance correctly as 5 units. However, the final assessment he made, "the point [tex]\((2, -2)\)[/tex] doesn't lie on the circle because the calculated distance should be the same as the radius," is contradictory to his correct distance calculation and the correct radius value.
Therefore, Amit's conclusion based on his own correct distance calculation was incorrect. The point [tex]\((2, -2)\)[/tex] does lie on the circle since the distance from the center to the point is equal to the radius.
