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SAT II Math I : Parabolas and Circles

Study concepts, example questions & explanations for SAT II Math I

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Example Questions

Example Question #132 : Geometry

Give the axis of symmetry of the parabola of the equation

$f(x) = 3x^{2} - 9x + 16$

Possible Answers:

$x = -\frac{3}{2}$

$x= \frac{1}{6}$

x = 16

$x= -\frac{1}{6}$

$x = \frac{3}{2}$

Correct answer:

$x = \frac{3}{2}$

Explanation:

The line of symmetry of the parabola of the equation

$f(x) = ax^{2}+ bx + c$

is the vertical line

$x = -\frac{b}{2a}$

Substitute a = 3, b = -9 :

The line of symmetry is

$x = -\frac{-9}{2 \cdot 3} = \frac{9}{6} = \frac{3}{2}$

That is, the line of the equation $x = \frac{3}{2}$ .

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Example Question #1 : Parabolas And Circles

What is the center of the circle with the following equation?

(x-7)^2+(y+12)^2=14

Possible Answers:

(0,14)

(7,14)

(7,-12)

(-7,12)

(14,7)

Correct answer:

(7,-12)

Explanation:

Remember that the basic form of the equation of a circle is:

(x-c_x)^2+(y-c_y)^2=r^2

This means that the center point (c_x,c_y) is defined by the two values subtracted in the squared terms. We could rewrite our equation as:

(x-(7))^2+(y-(-12))^2=14

Therefore, the center is (7,-12)

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Example Question #1 : Parabolas And Circles

What is the area of the sector of the circle formed between the -axis and the point on the circle found at x=6 when the equation of the circle is as follows?

x^2+y^2=100

Round your answer to the nearest hundreth.

Possible Answers:

91.24

71.42

314.16

46.37

11.14

Correct answer:

46.37

Explanation:

For this question, we will need to do three things:

Determine the point in question.
Use trigonometry to find the area of the angle in question.
Use the equation for finding a sector area to finalize our answer.

Let us first solve for the coordinate by substituting into our equation:

(6)^2+y^2=100

36+y^2=100

y^2=64

y=8

Our point is, therefore: (6,8)

Now, we need to calculate the angle formed between the origin and the point that we were given. We can do this using the inverse tangent function. The ratio of to is here: $\frac{8}{6}$

Therefore, the angle is:

$\tan^{-1}(\frac{8}{6})=53.13010235415592$

To solve for the sector area, we merely need to use our standard geometry equation. Note that the radius of the circle, based on the equation, is .

$\frac{53.13010235415592}{360}*\pi*10^2 = 46.36476090008$

This rounds to 46.37 .

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Example Question #1 : Parabolas And Circles

What is the area of the sector of the circle formed between the -axis and the point on the circle found at x=12.4 when the equation of the circle is as follows?

x^2+y^2=271

Round your answer to the nearest hundreth.

Possible Answers:

103.14

12.38

97.26

851.37

313.11

Correct answer:

97.26

Explanation:

For this question, we will need to do three things:

Determine the point in question.
Use trigonometry to find the area of the angle in question.
Use the equation for finding a sector area to finalize our answer.

Let us first solve for the coordinate by substituting into our equation:

12.4^2+y^2=271

153.76+y^2=271

y^2=117.24

y=10.82774214691133

Our point is, therefore: (12.4,10.82774214691133)

Now, we need to calculate the angle formed between the origin and the point that we were given. We can do this using the inverse tangent function. The ratio of to is here: $\frac{10.82774214691133}{12.4}$

Therefore, the angle is:

$\tan^{-1}(\frac{10.82774214691133}{12.4})=41.1276246899636$

To solve for the sector area, we merely need to use our standard geometry equation. Note that r^2 , based on the equation, is 271 .

$\frac{41.1276246899636}{360}*\pi*271 = 97.2635889213762$

This rounds to 97.26 .

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Example Question #5 : Parabolas And Circles

If the center of a circle is at $\left ( 0,0\right )$ and it has a radius of , what positive point on the y-axis does it intersect?

Possible Answers:

$\left ( 4,0\right )$

$\left ( 0,-4\right )$

$\left ( -4,0\right )$

$\left ( 0,4\right )$

Correct answer:

$\left ( 0,4\right )$

Explanation:

Since you are looking for a point on the y-axis , your value will be zero.

The center of the circle is at the origin and radius is the distance from the center, so that means the point you are looking for must be points away from .

This can be two points on the y-axis but since you are looking for a positive one, your answer must be $\left ( 0,4\right )$ .

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SAT II Math I : Parabolas and Circles

Study concepts, example questions & explanations for SAT II Math I

All SAT II Math I Resources

Example Questions

Example Question #132 : Geometry

Example Question #1 : Parabolas And Circles

Example Question #1 : Parabolas And Circles

Example Question #1 : Parabolas And Circles

Example Question #5 : Parabolas And Circles

All SAT II Math I Resources

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