A circle passes through points $A(1,2)$ and $B(3,4)$, and its center lies on the line $y=2 x-1$. What is the equation of this circle?
Options:
1. $(x-2)^2+(y-3)^2=2$
2. $(x-3)^2+(y-5)^2=10$
3. $(x-1)^2+(y-1)^2=8$
4. $(x-4)^2+(y-7)^2=26$
5. $x^2+(y+1)^2=10
Answer (2)
Find the center ( h , k ) of the circle using the given line equation y = 2 x − 1 , so k = 2 h − 1 .
Use the distance formula to set up an equation based on the fact that the distances from the center to points A ( 1 , 2 ) and B ( 3 , 4 ) are equal: ( h − 1 ) 2 + ( k − 2 ) 2 = ( h − 3 ) 2 + ( k − 4 ) 2 .
Substitute k = 2 h − 1 into the distance equation and solve for h , then find k .
Calculate the radius squared r 2 and write the equation of the circle: ( x − 2 ) 2 + ( y − 3 ) 2 = 2 .
Explanation
Problem Analysis We are given two points, A ( 1 , 2 ) and B ( 3 , 4 ) , that lie on a circle. The center of the circle lies on the line y = 2 x − 1 . We need to find the equation of this circle.
Setting up the equations Let the center of the circle be ( h , k ) . Since the center lies on the line y = 2 x − 1 , we have k = 2 h − 1 . The distance from the center to point A is equal to the distance from the center to point B (both are equal to the radius). Therefore, ( h − 1 ) 2 + ( k − 2 ) 2 = ( h − 3 ) 2 + ( k − 4 ) 2 .
Solving for h Substitute k = 2 h − 1 into the equation ( h − 1 ) 2 + ( k − 2 ) 2 = ( h − 3 ) 2 + ( k − 4 ) 2 :
( h − 1 ) 2 + ( 2 h − 1 − 2 ) 2 = ( h − 3 ) 2 + ( 2 h − 1 − 4 ) 2 ( h − 1 ) 2 + ( 2 h − 3 ) 2 = ( h − 3 ) 2 + ( 2 h − 5 ) 2 h 2 − 2 h + 1 + 4 h 2 − 12 h + 9 = h 2 − 6 h + 9 + 4 h 2 − 20 h + 25 5 h 2 − 14 h + 10 = 5 h 2 − 26 h + 34 − 14 h + 10 = − 26 h + 34 12 h = 24 h = 2
Solving for k Now, calculate k using k = 2 h − 1 :
k = 2 ∗ 2 − 1 k = 4 − 1 k = 3 So, the center of the circle is ( 2 , 3 ) .
Calculating the radius squared Calculate the radius squared, r 2 = ( h − 1 ) 2 + ( k − 2 ) 2 :
r 2 = ( 2 − 1 ) 2 + ( 3 − 2 ) 2 r 2 = 1 2 + 1 2 r 2 = 1 + 1 r 2 = 2
Finding the equation of the circle The equation of the circle is ( x − h ) 2 + ( y − k ) 2 = r 2 , so the equation is ( x − 2 ) 2 + ( y − 3 ) 2 = 2 .
Final Answer The equation of the circle is ( x − 2 ) 2 + ( y − 3 ) 2 = 2 .
Examples
Understanding the equation of a circle is crucial in various fields, such as engineering and computer graphics. For example, when designing a circular fountain, engineers need to determine the placement of the water jets to ensure they are equidistant from the center, creating a perfect circular spray pattern. Similarly, in computer graphics, circles are fundamental elements in creating images and animations, and knowing their equations allows developers to accurately render these shapes on the screen.
The equation of the circle that passes through points A(1,2) and B(3,4) with its center on the line y=2x-1 is ( x − 2 ) 2 + ( y − 3 ) 2 = 2 . By calculating the center and radius, we confirmed this result step-by-step. Therefore, the selected option is: (1) (x-2)^2+(y-3)^2=2.
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