Which equation represents a circle that contains the point $(-5,-3)$ and has a center at $(-2,1)$?
Distance formula: $\sqrt{\left(x_2-x_1\right)^2+\left(y_2-y_1\right)^2}$
A. $(x-1)^2+(y+2)^2=25$
B. $(x+2)^2+(y-1)^2=5$
C. $(x+2)^2+(y-1)^2=25$
D. $(x-1)^2+(y+2)^2=5
Answer (1)
Find the radius r using the distance formula between the center ( − 2 , 1 ) and the point ( − 5 , − 3 ) : r = 5 .
Write the equation of the circle with center ( − 2 , 1 ) and radius 5 : ( x − ( − 2 ) ) 2 + ( y − 1 ) 2 = 5 2 .
Simplify the equation: ( x + 2 ) 2 + ( y − 1 ) 2 = 25 .
The equation of the circle is: ( x + 2 ) 2 + ( y − 1 ) 2 = 25 .
Explanation
Problem Analysis The problem asks us to find the equation of a circle given its center and a point that lies on the circle. The general equation of a circle with center ( h , k ) and radius r is given by ( x − h ) 2 + ( y − k ) 2 = r 2 . We are given the center ( − 2 , 1 ) and a point on the circle ( − 5 , − 3 ) .
Find the radius First, we need to find the radius of the circle. The radius is the distance between the center and any point on the circle. We can use the distance formula to find the radius r between the center ( − 2 , 1 ) and the point ( − 5 , − 3 ) :
r = (( − 5 ) − ( − 2 ) ) 2 + (( − 3 ) − 1 ) 2
Calculate the radius Now, we simplify the expression for r :
r = ( − 3 ) 2 + ( − 4 ) 2 = 9 + 16 = 25 = 5
So, the radius of the circle is 5 .
Write the equation of the circle Now that we have the center ( − 2 , 1 ) and the radius r = 5 , we can write the equation of the circle:
( x − ( − 2 ) ) 2 + ( y − 1 ) 2 = 5 2
Simplifying, we get:
( x + 2 ) 2 + ( y − 1 ) 2 = 25
Final Answer Comparing this equation with the given options, we find that the correct equation is ( x + 2 ) 2 + ( y − 1 ) 2 = 25 .
Examples
Circles are fundamental in many real-world applications, from designing gears and wheels in mechanical engineering to modeling the orbits of planets in astronomy. For instance, if you're designing a circular garden and know the center's location and how far the edge should be from a specific point, you can use the equation of a circle to map out the garden's boundary accurately. This ensures your design fits perfectly within the available space and meets your aesthetic requirements.
