Finding Equation, Circle - Given 3 Points

Category: Analytic Geometry, Algebra

"Published in Newark, California, USA"

Find the equation of a circle that passes through the points (2, 3), (6, 1), and (4, -3).

Solution:

To illustrate the problem, it is better to sketch the circle with three points as follows:

Photo by Math Principles in Everyday Life

We know that the equation of a circle in standard form is 

                        (x - h)² + (y - k)² = r² 

where (h, k) is the coordinate of the center of a circle and r is the radius of a circle. In this problem, the center and radius of a circle are not given and we cannot use the equation of a circle in standard form. 

The equation of a circle in general form is 

                      x² + y² + Dx + Ey + F = 0

where D, E, and F are the coefficients. Since the three points of a circle are given, we can use the equation of a circle in general form to solve for D, E, and F. 

For (2, 3), if x = 2 and y = 3, the equation becomes

                 (2)² + (3)² + D(2) + E(3) + F = 0

                  4 + 9 + 2D + 3E + F = 0

                  2D + 3E + F = -13    (equation 1)    

For (6, 1), if x = 6 and y = 1, the equation becomes

                 (6)² + (1)² + D(6) + E(1) + F = 0

                 36 + 1 + 6D + E + F = 0

                 6D + E + F = -37      (equation 2) 

For (4, -3), if x = 4 and y = -3, the equation becomes

                 (4)² + (-3)² + D(4) + E(-3) + F = 0

                 16 + 9 + 4D - 3E + F = 0

                  4D - 3E + F = -25    (equation 3)

Since the three points of a circle are given, there are three equations and three unknowns. We can now solve for D, E, and F by elimination method.

If you subtract equation 2 from equation 1, 

 2D + 3E + F = -13                 2D + 3E + F = -13
                              --------->
-(6D + E + F = -37)                   -6D - E - F = 37
                                            ---------------------------
                                                  -4D + 2E = 24

                                                    -2D + E = 12     (equation 4)

If you subtract equation 3 from equation 2, 

   6D + E + F = -37                  6D + E + F = -37
                               --------->
-(4D - 3E + F = -25)              -4D + 3E - F = 25
                                           ----------------------------
                                                  2D + 4E = - 12    (equation 5)

Add equation 4 and equation 5,

     -2D + E = 12
    2D + 4E = -12
  ----------------------
            5E = 0

              E = 0

Substitute the value of E to equation 4 or equation 5,

    2D + 4E = -12
  2D + 4(0) = -12
            2D = -12

              D = -6

Substitute the value of D and E to equation 1, equation 2, or equation 3,

         2D + 3E + F = -13
    2(-6) + 3(0) + F = -13
                -12 + F = -13  

                         F = -1 

Substitute the value of D, E, and F to the equation of a circle in general form,

                             x² + y² + Dx + Ey + F = 0

                     x² + y² + (-6)x + (0)y + (-1) = 0

Therefore, the equation of a circle is

                             x² + y² - 6x - 1 = 0                                                     

                  

Proving Trigonometric Identities - Inverse Function

Category: Trigonometry

"Published in Newark, California, USA"

Prove the trigonometric identity for

Solution:

The first thing that you have to do is to examine the both sides of the equation and look for the more complicated side. In this case, the left side of the equation is more complicated. Let's simplify the left side of the equation as follows

The left side of the equation is the sum of two angles of tangent function. Let's apply the sum of two angles formula for tangent function as follows

Since each trigonometric functions at each term of the equation are inverse functions to each other, then we can cancel the tangent and inverse tangent functions at each term as follows

Therefore, 

Multiplying Both Two - Three Digit Numbers

Category: Arithmetic

"Published in Newark, California, USA"

How do you multiply these numbers in a shorter way without using a calculator: 
                        
                     66 x 64

Solution:

Well, we have the technique for that. What is the next number after 6? 7, isn't it? Therefore, 6 x 7 = 42. 42 will be the first two digits of the number.

                     4 2 _ _

How do you get the next two digits? Well, multiply both the one's digits in order to get the next two digits. Therefore, 6 x 4 = 24. 24 will be the next two digits of the number. The answer is

                     4 2 2 4

Note: There are two conditions in order for you to use this technique:

1. The ten's digits must be the same. In this case, both are starting with 6. 

2. The sum of the two one's digits must be equal to ten. In this case, 6 + 4 = 10.

Now, you know this technique. Let's try another one:

                     21 x 29

Solution:

The first two digits will be 2 x 3 = 6. Since the product is less than 10, the first digit will be 6. The next two digits will be 1 x 9 = 9. Since the product is less than 10, the next two digits must be written as 09. The answer is

                     6 0 9

Can we do this for both three digit numbers? Yes, we can do this also for both three digit numbers as long as the two conditions are followed. Let's try this one:

                   112 x 118

Solution:

As you notice that both numbers are starting with 11. The first digits will be 11 x 12 = 132. To multiply any two digit number by 11, place the first digit as the first digit of the product, place the last digit as the last digit of the product, and the middle digit of the product will be the sum of the two digits. In this case, the middle digit is 1 + 2 = 3. If the sum of the digits is 10 or more, use the ten's digit as a carry over to the first digit. 

The next two digits will be 2 x 8 = 16. Therefore, the answer is 

                    1 3 2 1 6