D AY 128 – E QUATION OF A CIRCLE CENTERED AT THE ORIGIN
I NTRODUCTION Circles are among the most common geometric figures we encounter in geometry. A circle can be drawn on the Cartesian plane; and coordinates used to describe the position of its center. The coordinates of the center of a circle and the length of its radius can be used to come up with an equation that represents that circle. We can also interpret and use the equation to find the center and radius of the circle. In this lesson, we will derive the equation of a circle centered at the origin using the Pythagorean theorem and also learn how to find the equation of a circle centered at the origin when the center and radius is given.
VOCABULARY Circles It is the path that is traced when a point moves around a fixed point while maintaining its distance from the point
The Pythagorean theorem is used when deriving the equation of a circle. Equally important is the distance formula. We apply the distance formula to find the length of the sides that form the right triangle which in turn leads to the derivation of the equation.
T HE EQUATION OF A CIRCLE CENTERED AT THE ORIGIN , 0, 0 Consider a circle with radius, OP = 𝑠 whose center is at the origin, O 0, 0 . 𝑦 P 𝑦, 𝑧 𝑠 𝑧 𝑧 Q O 0, 0 𝑦 , 0 𝑦
From the figure above; P 𝑦, 𝑧 represents any point on the circle. ∆POQ is right triangle with the right angle at Q and OP is the hypotenuse. Using the distance formula, we find out that: 𝑃𝑅 = 𝑦 and 𝑄𝑅 = 𝑧 Using the Pythagorean theorem, we have: OQ 2 + PQ 2 = OP 2 but OQ = 𝑦 and PQ = 𝑧 Therefore, we have: 𝒚 𝟑 + 𝒛 𝟑 = 𝒔 𝟑 This is the equation of a circle whose center is the origin 0, 0 and radius 𝑠 .
In order to find the equation of a circle, the center and radius of the circle must be known. Example 1 Find the equation of a circle with center 0, 0 and radius 7 units. Solution The center of the circle is the origin, therefore the equation takes the form 𝑦 2 + 𝑧 2 = 𝑠 2 with radius 7 units. The equation of the circle thus becomes: 𝑦 2 + 𝑧 2 = 7 2 ⇒ 𝒚 𝟑 + 𝒛 𝟑 = 𝟓𝟘
Example 2 Find the center and radius of a circle whose equation is 𝑦 2 + 𝑧 2 = 36 Solution The equation is in the form 𝑦 2 + 𝑧 2 = 𝑠 2 , therefore, definitely the center is 𝟏, 𝟏 . This means that 𝑠 2 = 36 ⇒ 𝑠 = 6 . The radius of the circle is 6 units.
Example 3 The equation of a circle is given by 2𝑦 2 + 2𝑧 2 = 18 . Determine the center and radius of the circle. Solution We rewrite the equation 2𝑦 2 + 2𝑧 2 = 18 in the form 𝑦 2 + 𝑧 2 = 𝑠 2 by dividing by each side of the equation by 2. The equation becomes: 𝑦 2 + 𝑧 2 = 9 . This shows that the center is 0, 0 and the radius is given by 𝑠 2 = 9 ⇒ 𝑠 = 9 = 3 units.
HOMEWORK A circle of radius 10 units has its center at the origin. Write down the equation of this circle.
A NSWERS TO HOMEWORK 𝑦 2 + 𝑧 2 = 100
THE END
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