Tuesday, July 15, 2014

Analytical Geometry : Circle Equations

Circle Equations from Math is Fun 

How to Find Equation of a Circle Passing 3 Given Points 

7 methods included ; Amazing !!

Practice finding the equation of a Circle given 3 points -- 

Q #1 : (1, 3), (7, 3) and (1, -3)

Answer : (x -4)2 + y2 = 18
Q #2  : (3, 4), (3, -4), (0, 5)
Answer : x2 + y2 = 25
Q #3 : A (1, 1), B (2, 4), C (5, 3)
Answer : (x-3)2 + (y -2)2 = 5
Solution : 
The midpoint of line AB on the Cartesian plane is \((\frac{3}{2}, \frac{5}{2})\) and the slope is \((\frac{3}{1})\) so the slope of the perpendicular bisector of line AB is \((\frac{-1}{3})\).
The equation of the line bisect line AB and perpendicular to line AB is thus :  
y - \((\frac{5}{2}\)) =\(\frac{-1}{3}\) [x - \((\frac{3}{2})\)] --- equation 1
The midpoint of line BC on the Cartesian plane is \((\frac{7}{2}, \frac{7}{2})\) and the slope is \((\frac{-1}{3})\) so the slope of the perpendicular bisector of line BC is 3.
And the equation of the line bisect line BC and perpendicular to line BC is 
y - \((\frac{7}{2})\) = 3 [x - \((\frac{7}{2})\)] --- equation 2
Solve the two equations for x and y and you have the center of the circle being (3, 2)
Use distance formula from the center circle to any point to get the radius = 
\(\sqrt{5}\).
The answer is : (x - 3)2 + (x - 2)2 = 5

More practices on similar questions :  (Answers below for self check)

Q #1 : A (2, 5) , B (2, 13) ,  and C (-6, 5 )

Q #2 : A (0, 7), B ( 6, 5 ), and C (-6, -11 )

Q #3 : A (3, -5) , B (-4, 2) and C (1, 7 )








Answer key :

#1 :  (x - 2) 2 + ( y - 9 ) 2 = 32

#2 :   x2 + (y + 3)2 = 100

#3 :  (x - 2) 2 + ( y -1) 2 = 37

Tuesday, July 8, 2014

This Week's Work : Week 53 for Inquisitive Young Mathletes

For this week's work, review 1991 Mathcounts National sprint and target round questions you got wrong or not fast enough. Please try once more to see if you now can solve them at ease.

For this week's review, try the following Mathcounts Mini :

Similar Triangles and Proportional Reasoning

Try the follow-up problems

Detailed solutions

I'll send you notes/solutions/links to some of the hardest problems later.

Take care and happy problem solving !!