To determine whether the point [tex]\((2, -2)\)[/tex] lies on the circle centered at [tex]\((-1, 2)\)[/tex] with a radius of 5 units, follow these steps:
1. Identify the center [tex]\((h, k)\)[/tex] of the circle and the given point [tex]\((x_1, y_1)\)[/tex]:
- Center of the circle: [tex]\((-1, 2)\)[/tex]
- Given point: [tex]\((2, -2)\)[/tex]
2. Recall the formula for the distance between two points [tex]\((x_1, y_1)\)[/tex] and [tex]\((x_2, y_2)\)[/tex]:
[tex]\[
d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}
\][/tex]
3. Substitute the given points into the distance formula:
- [tex]\(x_1 = -1\)[/tex]
- [tex]\(y_1 = 2\)[/tex]
- [tex]\(x_2 = 2\)[/tex]
- [tex]\(y_2 = -2\)[/tex]
[tex]\[
d = \sqrt{(2 - (-1))^2 + (-2 - 2)^2}
\][/tex]
Simplify the expression inside the square root:
[tex]\[
d = \sqrt{(2 + 1)^2 + (-2 - 2)^2}
\][/tex]
[tex]\[
d = \sqrt{3^2 + (-4)^2}
\][/tex]
Calculate the squares:
[tex]\[
d = \sqrt{9 + 16}
\][/tex]
[tex]\[
d = \sqrt{25}
\][/tex]
[tex]\[
d = 5
\][/tex]
4. Determine if the calculated distance is equal to the radius of the circle:
- The radius of the circle is 5 units.
- The calculated distance from the center [tex]\((-1, 2)\)[/tex] to the point [tex]\((2, -2)\)[/tex] is also 5 units.
Given that the distance [tex]\(d = 5\)[/tex] units is the same as the radius of the circle, this means the point [tex]\((2, -2)\)[/tex] indeed lies on the circle.
Therefore, Amit's conclusion is incorrect. He made a calculation mistake in determining the distance, but if calculated correctly, we see that the point [tex]\((2, -2)\)[/tex] is on the circle.
So, the right answer should be:
"No, he did not calculate the distance correctly."
