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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 Circle1)Given the equation of the line y = 7, derive the equation of a parallel line that passes through thepoint (-3,5). Show all steps involved in finding the equation in slope-intercept form.Answer:The slope-intercept equation of aline 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 thecircle 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 = r²General formula:x² + y²-2hx-2ky + h² + k²-r²= 0In this example we have the general formula :x² + y² +16x–18y+ 170 = 0then-2h =16,-2k =-18and h²+k²-r² = 76 therefore h =-8, k =9and r² =25 then r = 5The standard form of the circle is then :(x+8)² + (y-9)² =25The radius is : r =5The center is : (-8,9)The following diagram shows the plotted circle :

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