Asked by Umair Malik on Sep 23, 2024
Verified
Identify the center and radius of the circle x2−12x+y2−6y−36=0x ^ { 2 } - 12 x + y ^ { 2 } - 6 y - 36 = 0x2−12x+y2−6y−36=0 .
A) center: (12,6) radius: 216
B) center: (-6,-3) radius: 9
C) center: (6,3) radius: 36
D) center: (6,3) radius: 9
E) center: (12,6) radius: 36
Center
The midpoint of a geometric shape or object, or the average position of all the points in the shape.
Radius
The distance from the center of a circle to any point on its circumference, or from the center of a sphere to any point on its surface.
- Build capability in authoring and perceiving the traditional configurations of circle equations.
- Develop the ability to identify key features of conic sections such as center, radius, vertices, co-vertices, and foci.
Verified Answer
PS
Prabhjeet Singh3 days ago
Final Answer :
D
Explanation :
To identify the center and radius of a circle in standard form, we need to first complete the square for both the $x$ and $y$ terms.
$x^2 - 12x + y^2 - 6y - 36 = 0$
$x^2 - 12x + 36 + y^2 - 6y + 9 - 36 = 0$
$(x-6)^2 + (y-3)^2 = 36$
This is now in the form $(x-h)^2 + (y-k)^2 = r^2$, where the center is $(h,k)$ and the radius is $r$. So, we can identify the center as $(6,3)$ and the radius as $\sqrt{36} = 6$. Therefore, the best choice is D.
$x^2 - 12x + y^2 - 6y - 36 = 0$
$x^2 - 12x + 36 + y^2 - 6y + 9 - 36 = 0$
$(x-6)^2 + (y-3)^2 = 36$
This is now in the form $(x-h)^2 + (y-k)^2 = r^2$, where the center is $(h,k)$ and the radius is $r$. So, we can identify the center as $(6,3)$ and the radius as $\sqrt{36} = 6$. Therefore, the best choice is D.
Learning Objectives
- Build capability in authoring and perceiving the traditional configurations of circle equations.
- Develop the ability to identify key features of conic sections such as center, radius, vertices, co-vertices, and foci.