Graph of the System of Inequalities, 4

Category: Analytic Geometry

"Published in Vacaville, California, USA"

Graph the solution of the system of inequalities and find the coordinates of all vertices for

a. y < 9 - x²
    y ≥ x + 3

b. x² + y² ≤ 4
    x - y > 0

Solution:

a. For y < 9 - x², the given equation is a parabola that concave downward whose vertex is V(0, 9). Since the sign of inequality is less than, then all points along the curve are not included in the solution. If x = 0 and y = 0, then the given equation reduces to 0 < 9 which is correct and that point is included in the solution.

For y ≥ x + 3, the given equation is already written into slope-intercept form. Since the sign of inequality is greater than or equal to, then all points along the line are included in the solution. If x = 0 and y = 0, then the given equation reduces to 0 ≥ 3 which is correct and that point is included in the solution. Therefore, the graph of a pair of inequalities is

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The vertices of the graph are (-3, 0) and (2, 5) that are located at the intersection of the two shaded regions bounded by a line and a parabola. They are also included in the solution.

b. For x² + y² ≤ 4, the given equation is a circle whose center is C(0, 0) and radius is 2. Since the sign of inequality is less than or equal to, then all points along the curve are included in the solution. If x = 0 and y = 0, then the given equation reduces to 0 ≤ 4 which is correct and that point is included in the solution.

For x - y > 0, the given equation is already written into slope-intercept form. Since the sign of inequality is greater than, then all points along the line are not included in the solution. If x = 0 and y = 0, then the given equation reduces to 0 > 0 which is not correct and that point is not included in the solution. Therefore, the graph of a pair of inequalities is
 

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The vertices of the graph which are points A and B that are located at the intersection of the two shaded regions bounded by a line and a circle are also included in the solution.

Finding Equation - Circle

Category: Analytic Geometry, Plane Geometry, Algebra

"Published in Newark, California, USA"

Find the equation of a circle that is tangent to both axes, center in second quadrant and radius is 2.

Solution:

To illustrate the problem, it is better to draw the figure as follows

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When you say tangent, it is perpendicular. In this case the radius of a circle is perpendicular to x-axis and y-axis. If a circle is located at the second quadrant, the center is C (-2, 2). Therefore, the equation of a circle is

Circle and Secant Segment Problems, 4

Category: Plane Geometry

"Published in Newark, California, USA"

A circle can be drawn through points X, Y, and Z. 
a. What is the radius of the circle?
b. How far is the center of the circle from point W?

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Solution:

Consider the given figure above

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Expand the line segment YW from point W and label the other point as point V.

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If a theorem says "When two chords intersect inside a circle, the product of the segments of one chord equals the product of the segments of the other chord.", then the working equation is

Hence, the length of XZ is 20 and YV is 22. The midpoint of XZ is 10 units from X and the midpoint of YV is 11 units from Y. From their midpoints, draw a vertical and a horizontal line to locate their intersection which is point C as the center of a circle as follows

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From the figure, by using Pythagorean Theorem, the distance of C from W is

and the radius of a circle which is the distance of C from Z is

 

 

 

 

If you will get the distance of C from X, Y, or V, the value of length must be the same otherwise point C is not the center of a circle.

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Rotation of a Parabola, 2

Category: Analytic Geometry

"Published in Newark, California, USA"

Given the equation of a parabola

Find the new equation of a parabola if the given parabola is rotated counterclockwise about the origin at. 

Solution:

To illustrate the given problem, it is better to draw the figure as follows

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Since the given angle of rotation is written as inverse tangent function, then we can get the sine and cosine of the given angle of rotation by using basic trigonometric functions of a right triangle. 

 

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If the given equation is written in rectangular coordinate system, then we need to convert it into polar coordinate system as follows  

  

Next, substitute θ with θ - ϕ and then expand using the sum and difference of two angles formula, we have 

    
Convert the above equation into rectangular coordinate system in order to get its final equation. Therefore, the new equation of a parabola is 

Note: If the axes of any conic sections are not parallel to x and y axes, then the equation of any conic sections has xy term which is the general equation of any conic sections like parabola, ellipse, and hyperbola. Circle has no xy term always in any cases.