SAT II Math I : Coordinate Geometry

Example Questions

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Example Question #6 : How To Graph An Ordered Pair

Which of the following points is in Quadrant IV on the coordinate plane?

Two of these points are in Quadrant IV.

Explanation:

Quadrant IV consists of the points with positive -coordinates and negative -coordinates. Therefore is the correct choice.

Example Question #1 : Parabolas And Circles

Give the axis of symmetry of the parabola of the equation

Explanation:

The line of symmetry of the parabola of the equation

is the vertical line

Substitute :

The line of symmetry is

That is, the line of the equation .

Example Question #1 : Parabolas And Circles

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

Explanation:

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

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

Therefore, the center is

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  when the equation of the circle is as follows?

Explanation:

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

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

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

Our point is, therefore:

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:

Therefore, the angle is:

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 .

This rounds to .

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  when the equation of the circle is as follows?

Explanation:

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

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

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

Our point is, therefore:

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:

Therefore, the angle is:

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

This rounds to .

Example Question #5 : Parabolas And Circles

If the center of a circle is at  and it has a radius of , what positive point on the  does it intersect?

Explanation:

Since you are looking for a point on the , 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  but since you are looking for a positive one, your answer must be .

Example Question #1 : Coordinate Geometry

Given a point , what is the new  value if this point is flipped across the line ?

Explanation:

The displacement between negative three and positive one is four.

This mean that after flipping the point, it must be symmetrical to its original location.   The new point must also be 4 units to the right of the line.

The new point would be located at:

Example Question #2 : Symmetry

Which of the following symmetries applies to the graph of the relation

?

I) Symmetry with respect to the origin

II) Symmetry with respect to the -axis

III) Symmetry with respect to the -axis

None of these

II only

I, II and III

I only

III only

III only

Explanation:

The relation

is a circle with center  and radius  .

In other words, it is a circle with center at the origin, translated right  units and up  units (the radius is irrelevant to the question).

or

is this circle translated right zero units and up 2 units. The upshot is that the circle moves along the -axis only, and therefore is symmetric with respect to the -axis, but not the -axis. Also, as a consequence, it is not symmetric with respect to the origin.

Example Question #1 : Transformations

Reflect the point  across the line  and then across the origin.

Explanation:

Reflecting the point across the vertical line  will only change the x-value, but not the y-value.

The point after this reflection is:

Rotating this point across the origin will swap the x and y-values.

The new point is:

Example Question #1 : Coordinate Geometry

Let . If  is equal to  when flipped across the x-axis, what is the equation for ?