Question 11
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 (1)
Determine 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 points A ( 1 , 2 ) and B ( 3 , 4 ) lie on the circle.
Solve for h and k by substituting k = 2 h − 1 into the distance equation.
Find r 2 using the center ( h , k ) and one of the points, 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. We also know that the center of the circle lies on the line y = 2 x − 1 . Our goal is to find the equation of this circle.
Setting up the problem Let the center of the circle be ( h , k ) . Since the center lies on the line y = 2 x − 1 , we can write k = 2 h − 1 . The equation of a circle with center ( h , k ) and radius r is given by ( x − h ) 2 + ( y − k ) 2 = r 2 .
Using the distance formula Since points A ( 1 , 2 ) and B ( 3 , 4 ) lie on the circle, the distance from the center ( h , k ) to each of these points must be equal to the radius r . Therefore, we have:
( h − 1 ) 2 + ( k − 2 ) 2 = r 2 and ( h − 3 ) 2 + ( k − 4 ) 2 = r 2 .
This implies that ( h − 1 ) 2 + ( k − 2 ) 2 = ( h − 3 ) 2 + ( k − 4 ) 2 .
Solving for h Now, we 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 that we have h = 2 , we can find k using the equation k = 2 h − 1 :
k = 2 ( 2 ) − 1 = 4 − 1 = 3
So, the center of the circle is ( 2 , 3 ) .
Solving for r^2 Next, we find the radius squared, r 2 , using the distance formula with the center ( 2 , 3 ) and point A ( 1 , 2 ) :
r 2 = ( 2 − 1 ) 2 + ( 3 − 2 ) 2 = 1 2 + 1 2 = 1 + 1 = 2
Equation of the circle Finally, we can write the equation of the circle with center ( 2 , 3 ) and radius squared r 2 = 2 :
( 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 instance, when designing a circular garden or a roundabout, knowing the center and radius helps in accurately mapping out the structure. Similarly, in computer graphics, circles are fundamental elements, and their equations are used to draw and manipulate circular shapes on the screen. By grasping these concepts, you can apply them to real-world scenarios, creating precise and visually appealing designs.
