Determine the equation of a circle with a center at $(-4,0)$ that passes through the point $(-2,1)$.
Answer (1)
Calculate the radius r using the distance formula with the center ( − 4 , 0 ) and the point ( − 2 , 1 ) : r = ( − 2 − ( − 4 ) ) 2 + ( 1 − 0 ) 2 = 5 .
Substitute the center ( − 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: ( x − ( − 4 ) ) 2 + ( y − 0 ) 2 = ( 5 ) 2 .
Obtain the final equation of the circle: ( x + 4 ) 2 + y 2 = 5 .
Explanation
Problem Analysis The problem asks for the equation of a circle with a given center and a point it passes through. We'll use the distance formula to find the radius and then the standard equation of a circle to write the equation.
Calculate the Radius First, we need to find the radius of the circle. The center is ( − 4 , 0 ) and the circle passes through ( − 2 , 1 ) . We use the distance formula:
r = ( x 2 − x 1 ) 2 + ( y 2 − y 1 ) 2
Substituting the coordinates:
r = ( − 2 − ( − 4 ) ) 2 + ( 1 − 0 ) 2 r = ( 2 ) 2 + ( 1 ) 2 r = 4 + 1 r = 5
So, the radius is 5 .
Write the Equation of the Circle Now we use the standard form of the equation of a circle:
( 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, ( h , k ) = ( − 4 , 0 ) and r = 5 . Substituting these values, we get:
( x − ( − 4 ) ) 2 + ( y − 0 ) 2 = ( 5 ) 2 ( x + 4 ) 2 + y 2 = 5
Final Answer 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. For example, in architecture, arches and domes often utilize circular geometry for structural integrity and aesthetic appeal. Imagine designing a circular window for a building; determining its equation ensures precise construction and fitting. Similarly, in navigation, understanding circular paths is crucial for mapping routes and calculating distances, especially when dealing with satellite orbits or the range of radio signals.
