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QuestionFind the equations of a circle centered at (0,0)(0,0) with radius 33, given a point (0,4)(0,4) on it.

Studdy Solution

STEP 1

Assumptions1. The center of the circle is at the origin (0,0) . A point on the circle is (0,4)
3. The distance from the center to the point, which is the radius of the circle, is3 units

STEP 2

The standard equation of a circle with center at (h,k)(h,k) and radius rr is given by the formula(x−h)2+(y−k)2=r2(x-h)^2 + (y-k)^2 = r^2

STEP 3

Substitute the given values of the center (h,k)=(0,0)(h,k) = (0,0) and the radius r=3r =3 into the standard equation of the circle.
(x−0)2+(y−0)2=32(x-0)^2 + (y-0)^2 =3^2

STEP 4

implify the equation to obtain the standard equation of the circle.
x2+y2=9x^2 + y^2 =9

STEP 5

The general equation of a circle is given by the formulax2+y2+Dx+Ey+F=0x^2 + y^2 + Dx + Ey + F =0where =−2h = -2h, =−2k = -2k, and =h2+k2−r2 = h^2 + k^2 - r^2.

STEP 6

Substitute the given values of the center (h,k)=(0,0)(h,k) = (0,0) and the radius r=3r =3 into the formula for ,, , and $$.
=−2(0)=0 = -2(0) =0=−2(0)=0 = -2(0) =0=(0)2+(0)2−(3)2=−9 = (0)^2 + (0)^2 - (3)^2 = -9

STEP 7

Substitute the values of ,, , and $$ into the general equation of the circle.
x2+y2+0x+0y−9=0x^2 + y^2 +0x +0y -9 =0

STEP 8

implify the equation to obtain the general equation of the circle.
x2+y2−=0x^2 + y^2 - =0So, the standard equation of the circle is x2+y2=x^2 + y^2 = and the general equation of the circle is x2+y2−=0x^2 + y^2 - =0.

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