Write the standard equation of the circle with center $(5,7)$ and passes through the point $(2, 4)$.
A. $(x-5)^2+(y-7)^2=18$
B. $(x-5)^2+(y+7)^2=18$
C. $(x+5)^2+(y-7)^2=18$
D. $(x+5)^2+(y+7)^2=18
Answer (2)
Identify the center ( h , k ) of the circle as ( 5 , 7 ) .
Calculate the radius r using the distance formula between the center and the point ( 2 , 4 ) , finding r = 18 .
Determine r 2 = 18 .
Substitute the center and r 2 into the standard equation of a circle: ( x − 5 ) 2 + ( y − 7 ) 2 = 18 .
Explanation
Problem Analysis We are given the center of the circle as ( 5 , 7 ) and a point on the circle as ( 2 , 4 ) . Our goal is to find the standard equation of the circle.
Standard Equation of a Circle The standard equation of a circle with center ( h , k ) and radius r is given by: ( x − h ) 2 + ( y − k ) 2 = r 2
Substitute the Center In our case, the center is ( h , k ) = ( 5 , 7 ) . So, we have: ( x − 5 ) 2 + ( y − 7 ) 2 = r 2
Calculate the Radius To find the radius r , we use the distance formula between the center ( 5 , 7 ) and the point ( 2 , 4 ) on the circle: r = ( 2 − 5 ) 2 + ( 4 − 7 ) 2 r = ( − 3 ) 2 + ( − 3 ) 2 r = 9 + 9 r = 18 Therefore, r 2 = 18 .
Write the Final Equation Now, we substitute r 2 = 18 into the equation: ( x − 5 ) 2 + ( y − 7 ) 2 = 18
Final Answer The standard equation of the circle with center ( 5 , 7 ) and passing through the point ( 2 , 4 ) is: ( x − 5 ) 2 + ( y − 7 ) 2 = 18
Examples
Circles are fundamental in many real-world applications. For instance, in architecture, arches and domes often utilize circular geometry for structural integrity and aesthetic appeal. 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. Even in art and design, circles are used to create balanced and harmonious compositions, demonstrating their versatility and importance across various fields.
The standard equation of the circle with center ( 5 , 7 ) and passing through the point ( 2 , 4 ) is ( x − 5 ) 2 + ( y − 7 ) 2 = 18 . This is derived using the distance formula to find the radius and substituting into the standard equation format. The correct option is A.
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