A circle centered at [tex]$(-1,2)$[/tex] has a diameter of 10 units. Amit wants to determine whether [tex]$(2,-2)$[/tex] is also on the circle. His work is shown below.

The radius is 5 units.

Find the distance from the center to [tex]$(2,-2)$[/tex].
[tex]\[ \begin{array}{l}
\sqrt{(-1-2)^2+(2-(-2))^2} \\
\sqrt{(-3)^2+(4)^2}= \sqrt{9+16} = 5
\end{array} \][/tex]

The point [tex]$(2,-2)$[/tex] 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 [tex]$(2,-2)$[/tex] is the same as the radius.



Answer :

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."