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find all points using the distance formula











































6 units away from a point

Find all points on the y-axis (that is, having coordinates (0,y), where y is to be determined) that are 6 units distant from the point (4, -3).

Use the distance formula (Pythagorean theorem).

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Apr 03, 2014
6 units away from a point
by: Staff


Answer

Part I

The problem statement describes a circle whose center and radius are known

The standard equation for a circle is:

(x – circle center “x” coordinate)² + (y – circle center “y” coordinate)² = (radius of the circle)²

the center of the circle is known:

(4, -3)

at the center of the circle x = 4 and y = -3

the radius of the circle is also known:

radius = 6

therefore, the equation for the circle is:

(x – 4)² + (y – (-3))² = 6²



Circle with a radius of 6 units, whose center has the coordinates (4, -3).





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Apr 03, 2014
6 units away from a point
by: Staff


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Part II


the two points where the circle intersects the y-axis are the two solutions to the problem


The two points where the circle intersects the y-axis are the two solutions




Using the using the distance formula (Pythagorean theorem) to solve for side "b", the unknown side of the triangle.

The Pythagorean Theorem:


a² + b² = c²


substituting the known values

a = 4

c = 6


4² + b² = 6²



Using the distance formula (Pythagorean theorem) to solve for side 'b'







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Apr 03, 2014
6 units away from a point
by: Staff


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Part III


solve the equation for side "b"

4² + b² = 6²

4² - 4² + b² = 6² - 4²

0 + b² = 6² - 4²

b² = 6² - 4²

b² = 36 - 16

b² = 20

b = √(20)

b = 2√(5)

b ≈ 4.4721359549996


the two y values are:

y₁ (upper value) = y value for the point (0, -3) + b

y₁ (upper value) ≈ (-3) + 4.4721359549996

y₁ (upper value) ≈ 1.4721359549996

y₂ (lower value) = y value for the point (0, -3) - b

y₂ (lower value) ≈ (-3) - 4.4721359549996

y₂ (lower value) ≈ - 7.4721359549996



Coordinates of the two points that the circle intersects the y-axis.






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you can also solve for these values directly

use the equation for the circle:


(x – 4)² + (y – (-3))² = 6²


set x equal to 0


x = 0


(0 – 4)² + (y – (-3))² = 6²

(– 4)² + (y – (-3))² = 6²

(– 4)² + (y + 3)² = 6²

16 + y² + 6y + 9 = 36

y² + 6y + 9 + 16 = 36

y² + 6y + 25 = 36

y² + 6y + 25 - 36 = 36 - 36

y² + 6y - 11 = 0



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Apr 03, 2014
6 units away from a point
by: Staff


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Part IV



solve for y using the quadratic formula


The quadratic equation can also be used to calculate the position of the two points where the circle intersects the y-axis.





y² + 6y - 11 = 0

y = unknown

a = 1

b = 6

c = -11

Substitute the values of the coefficients a, b, and c into the quadratic formula


Substitute the values of the coefficients a, b, and c into the quadratic formula.






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Apr 03, 2014
6 units away from a point
by: Staff


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Part V


Calculate the final values of y using the quadratic formula.





Complete the arithmetic to find the two values of y








Thanks for writing.

Staff
www.solving-math-problems.com


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