GMAT Math : Calculating the equation of a circle

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Example Question #11 : Calculating The Equation Of A Circle

Two circles on the coordinate plane have the origin as their center. The outer circle has as its equation

$x^{2}+ y^2 = 64$ ;

the region between the circles has area $30 \pi$ .

Give the equation of the inner circle.

Possible Answers:

$x^{2}+ y^2 =34$

$x^{2}+ y^2 =17$

$x^{2}+ y^2 =49$

$x^{2}+ y^2 =30$

$x^{2}+ y^2 =4$

Correct answer:

$x^{2}+ y^2 =34$

Explanation:

A circle with its center at the origin has as its equation

$x^{2}+ y^2 = r^{2}$ .

Since the equation of the outer circle is $x^{2}+ y^2 = 64$ , then, for this circle,

$r^{2} = 64$ ,

and its area is $A = \pi r^{2} = \pi \cdot 64 = 64 \pi$ .

The area of the region between the circles is $30 \pi$ , so the inner circle has area

$64 \pi - 30 \pi = 34 \pi$ .

If is the radius of the inner circle, then its area is

$A = \pi R^{2} = 34\pi$

This makes $R ^{2} = 34$ , and the equation of the inner circle

$x^{2}+ y^2 = R^{2}$

$x^{2}+ y^2 =34$

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Example Question #12 : Calculating The Equation Of A Circle

Two circles on the coordinate plane have the origin as their center. The outer circle has twice the circumference as the inner circle, the equation of which is

$x^{2}+ y^{2} = 16$ .

Give the equation of the outer circle.

Possible Answers:

$x^{2}+ y^{2} = 18$

$x^{2}+ y^{2} = 32$

$x^{2}+ y^{2} = 20$

$x^{2}+ y^{2} = 64$

$x^{2}+ y^{2} = 128$

Correct answer:

$x^{2}+ y^{2} = 64$

Explanation:

The equation of a circle centered at the origin is

$x^{2}+ y^{2} = r^{2}$

where is the radius of the circle. Since the equation of the outer circle is

$x^{2}+ y^{2} = 16$ ,

$r^{2} = 16$ ,

and the radius is the square root of this:

r = 4 .

The circumference of this circle is $2 \pi$ times this, or

$C= 2 \pi \cdot 4 = 8 \pi$ .

The circumference of the larger circle is twice this, or $2 \cdot 8 \pi = 16 \pi$ ; divide this by $2 \pi$ to get the radius of the larger circle, which is

$R = 16 \pi \div 2 \pi = 8$ .

Consequently, $R ^{2} = 8 ^{2} = 64$ .

The equation of the larger circle is

$x^{2}+ y^{2} = R^{2}$ , or

$x^{2}+ y^{2} = 64$ .

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Example Question #13 : Calculating The Equation Of A Circle

Find the equation of circle whose radius is and center is at (-4,2) .

Possible Answers:

(x+4)^2+(y-2)^2=64

(x-4)^2+(y+2)^2=64

(x-4)^2+(y+2)^2=8

(x+4)^2+(y-2)^2=8

Correct answer:

(x+4)^2+(y-2)^2=64

Explanation:

To solve, simply use the formula given center at $\small (h, k)$ and radius of $\small r$ .

(x-h)^2+(y-k)^2=r^2

(x-(-4))^2+(y-(2))^2=8^2

(x+4)^2+(y-2)^2=64

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Example Question #14 : Calculating The Equation Of A Circle

Two circles on the coordinate plane have the origin as their center. The outer circle has twice the area as the inner circle, the equation of which is

$x^{2}+ y^{2} = 5$ .

Give the equation of the outer circle.

Possible Answers:

$x^{2}+ y^{2} = 5 \sqrt{2}$

$x^{2}+ y^{2} = 10$

$x^{2}+ y^{2} = 40$

$x^{2}+ y^{2} = 20$

$x^{2}+ y^{2} = 25$

Correct answer:

$x^{2}+ y^{2} = 10$

Explanation:

The equation of a circle centered at the origin is

$x^{2}+ y^{2} = r^{2}$

where is the radius of the circle. Since the equation of the outer circle is

$x^{2}+ y^{2} = 5$ ,

$r^{2} = 5$ ,

and the area is

$A = \pi r^{2} = \pi \cdot 5 = 5 \pi$ .

The area of the larger circle is twice this, or $2 \cdot 5 \pi = 10 \pi$ ; that is,

$A_{2} = \pi R^{2} = 10 \pi$

$R^{2} = 10$ ,

and the equation of that outer circle is

$x^{2}+ y^{2} = 10$ .

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Example Question #15 : Calculating The Equation Of A Circle

Two circles on the coordinate plane have the origin as their center. The outer circle has area five times that of the inner circle; the region between them has area $20 \pi$ . Give the equation of the inner circle.

Possible Answers:

$x^{2}+ y^2 = 4$

$x^{2}+ y^2 = 5$

$x^{2}+ y^2 =16$

$x^{2}+ y^2 = 8$

$x^{2}+ y^2 = 25$

Correct answer:

$x^{2}+ y^2 = 5$

Explanation:

Let be the radius of the inner circle. The area of the inner circle is $A = \pi r^{2}$ ; the outer circle has area five times this, or $A '= 5 \pi r^{2}$ ; the region between them has area equal to the difference of these quantities, or

$A'' = A ' - A= 5 \pi r^{2} - \pi r^{2} = 4 \pi r^{2}$

This is equal to $20 \pi$ , so

$4 \pi r^{2} = 20 \pi$

$\frac{4 \pi r^{2} }{4 \pi}= \frac{20 \pi}{4 \pi }$

$r^{2} =5$

A circle with its center at the origin has as its equation

$x^{2}+ y^2 = r^{2}$ ,

so the inner circle has as its equation

$x^{2}+ y^2 = 5$ .

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GMAT Math : Calculating the equation of a circle

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Example Question #12 : Calculating The Equation Of A Circle

Example Question #13 : Calculating The Equation Of A Circle

Example Question #14 : Calculating The Equation Of A Circle

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