Concentric Circles Problems, 2

Category: Plane Geometry

"Published in Vacaville, California, USA"

Find the area of the circular section as shown in the figure

Photo by Math Principles in Everyday Life

Solution: 

If a small circle is tangent to the chord of a big circle, then the chord is bisected at the point of tangency. The radii of two circles are perpendicular to the point of tangency. To analyze more the problem, it is better to label further the figure as follows

Photo by Math Principles in Everyday Life

Let R₁ = be the radius of a big circle
      R₂ = be the radius of a small circle

If you connect the half of a chord and radii of two circles, then it becomes a right triangle. By Pythagorean Theorem, we can have the first working equation as follows

The area of a big circle is  .

The area of a small circle is .

The area of the circular section is
 

 

 

But

Therefore, the area of the circular section is

 

 

                               or

 

 

 

Theory of Equations, 8

Category: Algebra

"Published in Newark, California, USA"

Find the equation of a polynomial if the roots are 3, 1, and -√2. 

Solution:

If one of the root of the equation is given which is -√2 , then we need to get its conjugate because we want to eliminate the radical sign in the given equation. The conjugate of -√2 is √2 . Hence, the equation or a factor from the product of a root and its conjugate is   

Therefore, the equation of a polynomial is

Theory of Equations, 7

Category: Algebra

"Published in Newark, California, USA"

Find the remaining roots of the equation

if 1 - i√2 is a root.

Solution:

If one of the root of the equation is given which is 1 - i√2 , then we need to get its conjugate because we want to eliminate the imaginary number and the radical sign in the given equation. The conjugate of 1 - i√2 is 1 + i√2 . Hence, the equation or a factor from the product of a root and its conjugate is   

In order to get the other factor for the given equation, let's divide the given equation with the above equation, we have 

     
Since there's no remainder in the division, then the other factor of the given equation is x² + 2x - 3.    

Let's factor the other factor of the given equation as follows

If you equate each factor to zero, then the values of x are -3 and 1.

Therefore, the other roots of the given equation are -3 and 1.

 

Theory of Equations, 6

Category: Algebra

"Published in Newark, California, USA"

Find the remaining roots of the equation

if 2 + √5 is a root.

Solution:

If one of the root of the equation is given which is 2 + √5 , then we need to get its conjugate because we want to eliminate the radical sign in the given equation. The conjugate of 2 + √5 is 2 - √5. Hence, the equation or a factor from the product of a root and its conjugate is  

In order to get the other factor for the given equation, let's divide the given equation with the above equation, we have  

Since there's no remainder in the division, then the other factor of the given equation is 2x + 3.   

Therefore, the other root of the given equation is .
 

Theory of Equations, 5

Category: Algebra

"Published in Newark, California, USA"

Find the remaining roots of the equation

if -√2 is a root.

Solution:

If one of the root of the equation is given which is -√2, then we need to get its conjugate because we want to eliminate the radical sign in the given equation. The conjugate of -√2 is √2. Hence, the equation or a factor from the product of a root and its conjugate is 

In order to get the other factor for the given equation, let's divide the given equation with the above equation, we have 

Since there's no remainder in the division, then the other factor of the given equation is x² - 2x + 4.  

By using the completing the square method, we can solve for the other roots of the given equation as follows

Therefore, the other roots of the given equation are and .

Theory of Equations, 4

Category: Algebra

"Published in Newark, California, USA"

Find the remaining roots of the equation

if 5 + i is a root.

Solution:

If one of the root of the equation is given which is 5 + i, then we need to get its conjugate because we want to eliminate the imaginary number in the given equation. The conjugate of 5 + i is 5 - i. Hence, the equation or a factor from the product of a root and its conjugate is

In order to get the other factor for the given equation, let's divide the given equation with the above equation, we have

Since there's no remainder in the division, then the other factor of the given equation is x + 2. 

Therefore, the other root of the given equation is - 2.