Use the image to answer the question.
A circle and 4 points are plotted on a coordinate plane with x-axis ranging from negative 7 to 7 in 1 unit increments and y-axis ranging from negative 9 to 4 in 1 unit increments.
Ricardo draws a circle on the coordinate plane. The circle has the equation (16−(x+2)2)−−−−−−−−−−−−−√−5
. The circle is now translated 3 units to the right and 3 units up. Where is the translated circle’s center?
(1 point)
(
,
)
The Answer is actually (1,-2)
The answer to question 3 is -2
To find the center of the translated circle, we need to identify the coordinates after the translation.
The original equation of the circle is (16 - (x + 2)^2) + √(-5). Let's consider the center of the circle in its original position as (h, k).
In the equation of the circle (x - h)^2 + (y - k)^2 = r^2, where (h, k) gives the coordinates of the center.
Since the circle's equation is given in the form (x + 2)^2 + (y + √(-5))^2 = 16, comparing the two equations, we can determine that the center of the original circle is (-2, -√(-5)).
Now, let's calculate the coordinates of the translated circle's center.
Given that the translation moves the circle 3 units to the right and 3 units up, we add 3 to the x-coordinate and 3 to the y-coordinate of the original center.
Thus, the coordinates of the translated circle's center are:
x-coordinate: -2 + 3 = 1
y-coordinate: -√(-5) + 3 = -√(-5) + 3
Therefore, the translated circle's center is (1, -√(-5) + 3).
To answer this question, we need to understand a few concepts and steps:
1. Equation of a Circle: The equation of a circle in standard form is (x - h)^2 + (y - k)^2 = r^2, where (h, k) represents the coordinates of the center of the circle, and r represents the radius.
2. Translation: When a shape is translated, its position is shifted horizontally and/or vertically without changing its shape or size. In this case, the original circle is translated 3 units to the right and 3 units up.
Now let's find the center of the translated circle:
1. Original Center: The equation of the original circle is (16 - (x + 2)^2)^0.5 - 5. From this equation, we can see that the original center of the circle is (-2, -5). This is because the equation is in the form (x - h)^2 + (y - k)^2 = r^2, where (h, k) is the center of the circle.
2. Translation: To translate the circle 3 units to the right and 3 units up, we need to add 3 to the x-coordinate and 3 to the y-coordinate of the original center.
Thus, the translated center would be (-2 + 3, -5 + 3) = (1, -2).
Therefore, the translated circle's center is (1, -2).