March 31st, 2018, 05:44 AM  #1 
Senior Member Joined: Nov 2010 From: Indonesia Posts: 2,001 Thanks: 132 Math Focus: Trigonometry  [ASK] Circle Equation
A circle L is going through the point O (0, 0) and P (6, 0). The center is in the line $\displaystyle y=\frac{4}{3}x$. The equation of the circle L is .... A. $\displaystyle x^2+y^2+6x8y=0$ B. $\displaystyle x^2+y^26x8y=0$ C. $\displaystyle x^2+y^28x6y=0$ D. $\displaystyle x^2+y^2+8x+6y=0$ E. $\displaystyle x^2+y^24x3y=0$ Since the equation of a circle is $\displaystyle x^2+y^2+Ax+By+C=0$, I substituted both known points to the equation and got C = 0 as well as B = 6, so the answer is obviously B. But then my student asked "What if all options have 6 as their B? How would we know the answer?". I think it has something to do with that $\displaystyle y=\frac{4}{3}x$, but how? Please give me some hints. 
March 31st, 2018, 09:34 AM  #2 
Senior Member Joined: Sep 2015 From: USA Posts: 2,530 Thanks: 1390 
let $C$ be the center. $\C(0,0)\^2 = \C  (6,0)\^2 $ $C_x^2 + C_y^2 = (C_x6)^2 + C_y^2$ $C_x^2 = (C_x6)^2$ $C_x = 6C_x$ $C_x = 3$ And clearly $C_y=4$ The radius is thus $r = \(3,4)\ = 5$ thus our circle is given by $(x3)^2 + (y4)^2 = 25$ which simplifies to $x^2  6x + y^2  8y = 0$ i.e answer (b) 
March 31st, 2018, 11:10 AM  #3 
Math Team Joined: Jan 2015 From: Alabama Posts: 3,264 Thanks: 902 
If (0, 0) and (6, 0) lie on the circle then the perpendicular bisector of the segment passes through the center of the circle. The line through the points (0, 0) and (6, 0) is, of course, the xaxis so the perpendicular bisector is parallel to the yaxis. Its equation is x= 3. The center of the circle also lies on the line y= (4/3)x. The center is x= 3, y= (4/3)(3)= 4, (3, 4). The distance from (0, 0) to (3, 4) is $\displaystyle \sqrt{9+ 16}= 5$. As a check the distance from (6, 0) to (3, 4) is $\displaystyle \sqrt{(6 3)^2+ 16}= \sqrt{9+ 16}= 5$ also. The equation of the circle is $\displaystyle (x 3)^2+ (y 4)^2= x^2 6x+ 9+ y^2 8y+ 16= 25$ which reduces to $\displaystyle x^2+ y^2 6x 8y= 0$. 
March 31st, 2018, 04:42 PM  #4 
Senior Member Joined: Nov 2010 From: Indonesia Posts: 2,001 Thanks: 132 Math Focus: Trigonometry 
Thanks for both of you!


Tags 
circle, equation 
Thread Tools  
Display Modes  

Similar Threads  
Thread  Thread Starter  Forum  Replies  Last Post 
Circle Equation  red87445  Geometry  4  April 26th, 2014 05:56 PM 
Equation of a Circle  bilano99  Algebra  8  May 16th, 2013 08:01 AM 
Equation of a circle  weatherg  Complex Analysis  3  October 9th, 2011 05:58 PM 
the equation of a circle  hailua  Algebra  1  April 28th, 2010 03:12 AM 
Circle equation  football  Algebra  1  April 8th, 2010 04:46 AM 