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 (2)
Calculate the radius r using the distance formula between the center ( − 4 , 0 ) and the point ( − 2 , 1 ) , resulting in r = 5 .
Substitute the center coordinates ( h , k ) = ( − 4 , 0 ) and the radius r = 5 into the standard equation of a circle ( x − h ) 2 + ( y − k ) 2 = r 2 .
Simplify the equation to ( x + 4 ) 2 + y 2 = 5 .
State the final equation of the circle: ( x + 4 ) 2 + y 2 = 5 .
Explanation
Problem Analysis The problem asks us to find the equation of a circle given its center and a point it passes through. We will use the distance formula to find the radius and then substitute the center and radius into the standard equation of a circle.
Calculate the Radius First, we need to find the radius of the circle. The radius is the distance between the center ( − 4 , 0 ) and the point ( − 2 , 1 ) on the circle. We use the distance formula: r = (( − 2 ) − ( − 4 ) ) 2 + ( 1 − 0 ) 2 r = ( 2 ) 2 + ( 1 ) 2 r = 4 + 1 r = 5
Write the Equation of the Circle Now that we have the radius r = 5 , we can find the equation of the circle. The standard form of a circle's equation is: ( x − h ) 2 + ( y − k ) 2 = r 2 where ( h , k ) is the center of the circle and r is the radius. In our case, the center is ( − 4 , 0 ) , so h = − 4 and k = 0 . The radius is 5 , so r 2 = ( 5 ) 2 = 5 . Substituting these values into the standard equation, we get: ( x − ( − 4 ) ) 2 + ( y − 0 ) 2 = ( 5 ) 2 ( x + 4 ) 2 + y 2 = 5
Final Answer Therefore, the equation of the circle with center ( − 4 , 0 ) that passes through the point ( − 2 , 1 ) is: ( x + 4 ) 2 + y 2 = 5
Examples
Circles are fundamental in many real-world applications, from designing wheels and gears to understanding planetary orbits. For instance, engineers use the equation of a circle to design circular gears in machinery, ensuring precise and efficient motion. Architects also use circles in building designs for aesthetic and structural purposes, such as domes and arches. Understanding the equation of a circle allows for precise calculations and designs in these diverse fields.
The equation of the circle with center at (-4,0) that passes through the point (-2,1) is (x + 4)^2 + y^2 = 5. This was determined by finding the radius using the distance formula and substituting into the circle's standard equation. The selected answer corresponds to option C.
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