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)
Find the relation between the center's coordinates ( h , k ) using the given line equation: k = 2 h − 1 .
Use the distance formula to equate the distances from the center to points A ( 1 , 2 ) and B ( 3 , 4 ) .
Solve for h and k , obtaining the center ( 2 , 3 ) .
Calculate r 2 and write the equation of the circle: ( x − 2 ) 2 + ( y − 3 ) 2 = 2 .
Explanation
Find the relation between h and k 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 .
Equalize distances The distance from the center ( h , k ) to point A ( 1 , 2 ) is equal to the distance from the center ( h , k ) to point B ( 3 , 4 ) . This distance is the radius r of the circle. Using the distance formula, we have ( h − 1 ) 2 + ( k − 2 ) 2 = ( h − 3 ) 2 + ( k − 4 ) 2
Solve 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
Solve for k Now, find k using k = 2 h − 1 :
k = 2 ( 2 ) − 1 = 4 − 1 = 3
Calculate radius squared The center of the circle is ( 2 , 3 ) . Calculate the radius r using the distance formula between the center ( 2 , 3 ) and 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 in the form ( x − h ) 2 + ( y − k ) 2 = r 2 . Substitute h = 2 , k = 3 , and 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. For instance, in GPS technology, determining your precise location relies on calculating the intersection points of circles from multiple satellites. Each satellite transmits a signal indicating its distance from your device, defining a circle around the satellite. The intersection of these circles pinpoints your location on Earth. Similarly, in medical imaging, the circular shape is fundamental in interpreting scans like CT scans and MRIs, where understanding the geometry helps doctors identify anomalies and diagnose conditions accurately.
