Examkin

Question

Difficulty: MediumEquations and Graphs of Circles

A circle graphed on the coordinate plane is represented by the equation x2+y2−12x+16y+19=0x^2 + y^2 - 12x + 16y + 19 = 0. What is the length of the radius of the circle?

Answer: 9

Answer

The radius of the circle is 9.
Converting the given general equation x2+y2−12x+16y+19=0x^2 + y^2 - 12x + 16y + 19 = 0 into standard form (x−h)2+(y−k)2=r2(x - h)^2 + (y - k)^2 = r^2 by completing the square yields (x−6)2+(y+8)2=81(x - 6)^2 + (y + 8)^2 = 81. Taking the square root of 81 gives a radius of 9.

Step-by-Step Solution

1
Rearrange the equation to group the x-terms and y-terms together and move the constant term to the right side.
(x2−12x)+(y2+16y)=−19(x^2 - 12x) + (y^2 + 16y) = -19
This sets up the expression for completing the square.
2
Complete the square for both the xx and yy variable groups by adding (12/2)2=36(12/2)^2 = 36 and (16/2)2=64(16/2)^2 = 64 to both sides of the equation.
(x−6)2+(y+8)2=−19+36+64=81(x - 6)^2 + (y + 8)^2 = -19 + 36 + 64 = 81
Adding these values balances the equation and converts the quadratic expressions into perfect square binomials.
3
Compare to the standard equation of a circle (x−h)2+(y−k)2=r2(x - h)^2 + (y - k)^2 = r^2 to determine the radius rr.
r2=81  ⟹  r=81=9r^2 = 81 \implies r = \sqrt{81} = 9
The constant term on the right side of the standard form equation equals the square of the radius.

Key Concept

Standard form equation of a circle and completing the square
Rate this question

Topics

Equations and Graphs of Circles

More questions from this topic