Deriving The Equation Of a Parallel Line And Standard Form Of a Circle

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Deriving the Equation of a Parallel Line and Standard Form of a Circle 1) Given the equation of the line y = 7, derive the equation of a parallel line that passes through the point ( - 3,5). Show all steps involved in finding the equation in slope - intercept form. Answer: The slope - intercept equation of a line is : y = mx + p. An equation parallel to the line y = 7 has the same slope, i.e. m = 0 : the equation of the line is then y = p. This line passes through the point ( - 3,5) then the y - intercept p is equal to 5. Therefore the equation of the line, in slope - intercept form, is : y = 5. (slope m =0, y - intercept p = 5) 2) Using the general equation of a circle, x² + y² + 16x – 18y + 170 = 0, derive the standard form of the circle equation. Also, calculate the radius and the center of the circle. Answer: For a circle of radius r and center (h,k) we have the two following formulas : Standard formula : (x - h)² + (y - k )² = r² General formula : x² + y² - 2hx - 2ky + h² + k² - r² = 0 In this example we have the general formula : x² + y² + 16 x – 18 y + 170 = 0 then - 2h = 16 , - 2k = - 1 8 and h²+k² - r² = 76 therefore h = - 8 , k = 9 and r² = 25 then r = 5 The standard form of the circle is then : (x+ 8 )² + (y - 9 )² = 25 The radius is : r = 5 The center is : ( - 8 , 9 ) The following diagram shows the plotted circle :
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