# 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).

### Comments for find all points using the distance formula

 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 --------------------------------------------

 Apr 03, 2014 6 units away from a point by: Staff -------------------------------------------- Part II the two points where the circle intersects the y-axis are the two solutions to the problem 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 --------------------------------------------

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

 Apr 03, 2014 6 units away from a point by: Staff -------------------------------------------- Part IV solve for y using the quadratic formula 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 --------------------------------------------

 Apr 03, 2014 6 units away from a point by: Staff -------------------------------------------- Part V Thanks for writing. Staff www.solving-math-problems.com

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