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HiSET: Math : Equations of circles

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Example Questions

Example Question #151 : Hi Set: High School Equivalency Test: Math

A circle on the coordinate plane has center (2,9 ) and passes through (10, 3) . Give its area.

Possible Answers:

$10 \pi$

$20 \pi$

$100 \pi$

$25 \pi$

The information given is insufficient to answer the question

Correct answer:

$100 \pi$

Explanation:

The radius of the circle is the distance between its center and a point that it passes through. This radius can be calculated using the distance formula:

$r = \sqrt{(x_{2}-x_{1})^{2}+(y_{2}-y_{1})^{2} }$ .

Setting $x_{1} = 2, x_{2} = 10, y_{1}=9, y_{2} =3$ :

$r = \sqrt{(10-2)^{2}+(3-9)^{2} }$

$= \sqrt{8^{2}+(-6)^{2} }$

$= \sqrt{64+36}$

$= \sqrt{100}$

=10

The area of a circle can be calculated by substituting 10 for in the equation

$A = \pi r^{2}$

$= \pi (10)^{2}$

$= 100 \pi$ ,

the correct response.

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Example Question #152 : Hi Set: High School Equivalency Test: Math

Give the area of the circle on the coordinate plane with equation

$x^{2}+y^{2}+ 6x-20y+ 9=0$ .

Possible Answers:

$20 \pi$

$40 \pi$

$100 \pi$

$10 \pi$

$50 \pi$

Correct answer:

$100 \pi$

Explanation:

We must first rewrite the equation of the circle in standard form

$(x-h)^{2} + (y-k)^{2}= r^{2}$ ;

will be the radius.

$x^{2}+y^{2}+ 6x-20y+ 9=0$

Subtract 9 from both sides, and rearrange the terms remaining on the left as follows:

$x^{2}+y^{2}+ 6x-20y+ 9- 9 =0- 9$

$x^{2}+ 6x + \underline{\; \; \; \; \; \; }+y^{2}-20y+ \underline{\; \; \; \; \; \; } = - 9$

Note that blanks have been inserted after the linear terms. In these blanks, complete two perfect square trinomials by dividing linear coefficients by 2 and squaring:

$(6 \div 2)^{2} = 3^{2} = 9$

$(20 \div 2)^{2} = 10^{2} = 100$

Add to both sides, as follows:

$x^{2}+ 6x + 9+y^{2}-20y+ 100 = - 9+9+ 100$

$x^{2}+ 6x + 9+y^{2}-20y+ 100 = 100$

Rewrite the expression on the left as the sum of the squares of two binomials.

$(x+3)^{2} + (y-10)^{2} = 100$

The equation is now in standard form.

$r^{2} = 100$

The area of this circle can be found using this formula:

$A = \pi r^{2} = \pi (100) = 100 \pi$ ,

the correct response.

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HiSET: Math : Equations of circles

Study concepts, example questions & explanations for HiSET: Math

All HiSET: Math Resources

Example Questions

Example Question #151 : Hi Set: High School Equivalency Test: Math

Example Question #152 : Hi Set: High School Equivalency Test: Math

All HiSET: Math Resources

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