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)
• Express the center of the circle in terms of a single variable using the given line equation. • Use the distance formula to equate the distances from the center to the two points on the circle. • Solve for the coordinates of the center and then find the radius squared. • Write the equation of the circle using the center and radius: ( 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 . Our goal is to find the equation of this circle.
Express k in terms of h Let the center of the circle be ( h , k ) . Since the center lies on the line y = 2 x − 1 , we can express k in terms of h as k = 2 h − 1 .
Distance Formula The distance from the center ( h , k ) to point A ( 1 , 2 ) must be equal to the distance from the center ( h , k ) to point B ( 3 , 4 ) because both points lie on the circle. We can use the distance formula to express this equality: ( h − 1 ) 2 + ( k − 2 ) 2 = ( h − 3 ) 2 + ( k − 4 ) 2 Squaring both sides to eliminate the square roots, we get: ( h − 1 ) 2 + ( k − 2 ) 2 = ( h − 3 ) 2 + ( k − 4 ) 2
Substitute k = 2h - 1 Now, substitute k = 2 h − 1 into the equation: ( 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
Solve for h Expand and simplify the equation: 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
Solve for k Now that we have h = 2 , we can find k using k = 2 h − 1 :
k = 2 ( 2 ) − 1 = 4 − 1 = 3
Calculate radius squared So, the center of the circle is ( 2 , 3 ) . Now we need to find the radius squared, r 2 . We can use the distance formula from the center to either point A or B. Let's use point A(1, 2): r 2 = ( 2 − 1 ) 2 + ( 3 − 2 ) 2 = 1 2 + 1 2 = 1 + 1 = 2
Write the equation of the circle The equation of the circle is ( x − h ) 2 + ( y − k ) 2 = r 2 , so substituting the values we found: ( x − 2 ) 2 + ( y − 3 ) 2 = 2
Final Answer Comparing this with the given options, we see that it matches option 1.
Examples
Understanding the equation of a circle is crucial in various fields, such as GPS navigation. Imagine you're using a GPS app on your phone. The app uses signals from satellites to determine your location. Each satellite's signal can be thought of as defining a circle (or sphere in 3D) centered at the satellite's position, with the radius being the distance from the satellite to your phone. By finding the intersection of these circles (or spheres), the app can pinpoint your exact location on Earth. This is a direct application of solving for the equation of a circle and understanding its properties.
