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Math  /  Geometry

QuestionGive the equation of the circle centered at the origin and passing through the point (0,−4)(0,-4). □\square

Studdy Solution

STEP 1

1. A circle is defined as the set of all points equidistant from a fixed point, called the center.
2. The general equation of a circle centered at the origin (0,0)(0,0) is given by x2+y2=r2x^2 + y^2 = r^2, where rr is the radius of the circle.
3. The radius rr can be found using the distance formula if a point on the circle is known.

STEP 2

1. Determine the radius of the circle.
2. Write the equation of the circle using the radius.

STEP 3

Determine the radius rr of the circle using the given point (0,−4)(0, -4).
The radius rr is the distance from the center (0,0)(0,0) to the point (0,−4)(0, -4).
Using the distance formula: r=(0−0)2+(−4−0)2=0+16=16=4 r = \sqrt{(0-0)^2 + (-4-0)^2} = \sqrt{0 + 16} = \sqrt{16} = 4

STEP 4

Write the equation of the circle using the radius r=4r = 4.
The general equation of a circle centered at the origin (0,0)(0,0) is: x2+y2=r2 x^2 + y^2 = r^2
Substituting r=4r = 4: x2+y2=42 x^2 + y^2 = 4^2 x2+y2=16 x^2 + y^2 = 16
Thus, the equation of the circle centered at the origin and passing through the point (0,−4)(0, -4) is: x2+y2=16 x^2 + y^2 = 16

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