Determine the equation of a circle with a center at $(-4,0)$ that passes through the point $(-2,1)$ by following the steps below.
1. Use the distance formula to determine the radius: $d=\sqrt{\left(x_2-x_1\right)^2+\left(y_2-y_1\right)^2}$.
2. Substitute the known values into the standard form: $(x-h)^2+(y-k)^2=r^2$.
What is the equation of a circle with a center at $(-4,0)$ that passes through the point $(-2,1)$?
A. $x^2+(y+4)^2=\sqrt{5}$
B. $(x-1)^2+(y+2)^2=5$
C. $(x+4)^2+y^2=5$
D. $(x+2)^2+(y-1)^2=\sqrt{5}$
Answer (1)
Calculate the radius r using the distance formula: r = (( − 2 ) − ( − 4 ) ) 2 + ( 1 − 0 ) 2 = 5 .
Determine r 2 : r 2 = ( 5 ) 2 = 5 .
Substitute the center ( − 4 , 0 ) and r 2 = 5 into the standard circle equation: ( x − ( − 4 ) ) 2 + ( y − 0 ) 2 = 5 .
Simplify the equation to get the final answer: ( x + 4 ) 2 + y 2 = 5 .
Explanation
Problem Analysis We are given a circle with center ( − 4 , 0 ) that passes through the point ( − 2 , 1 ) . We need to find the equation of this circle. The standard form of a circle's equation is ( x − h ) 2 + ( y − k ) 2 = r 2 , where ( h , k ) is the center and r is the radius.
Calculate the Radius First, we need to find the radius of the circle. We can use the distance formula to find the distance between the center ( − 4 , 0 ) and the point ( − 2 , 1 ) on the circle. The distance formula is d = ( x 2 − x 1 ) 2 + ( y 2 − y 1 ) 2 . In our case, ( x 1 , y 1 ) = ( − 4 , 0 ) and ( x 2 , y 2 ) = ( − 2 , 1 ) . So, the radius r is: r = (( − 2 ) − ( − 4 ) ) 2 + ( 1 − 0 ) 2 r = ( 2 ) 2 + ( 1 ) 2 r = 4 + 1 r = 5
Calculate r^2 Now that we have the radius r = 5 , we can find r 2 :
r 2 = ( 5 ) 2 = 5
Write the Equation of the Circle Next, we substitute the center ( h , k ) = ( − 4 , 0 ) and r 2 = 5 into the standard equation of a circle: ( x − ( − 4 ) ) 2 + ( y − 0 ) 2 = 5 ( x + 4 ) 2 + y 2 = 5
Final Answer Therefore, the equation of the circle is ( x + 4 ) 2 + y 2 = 5 .
Examples
Circles are fundamental in many real-world applications, from designing gears and wheels to understanding planetary orbits. For instance, engineers use the equation of a circle to ensure that a circular gear of a specific radius will properly mesh with other gears in a machine. Similarly, architects use circular geometry to design domes and arches, ensuring structural integrity and aesthetic appeal. Understanding the equation of a circle allows us to model and analyze these shapes effectively.
