Coordinate Geometry - GMAT Quantitative Reasoning

Find endpoint Y given the following:

I) Segment RY has its midpoint at (45,65)

II) Point R is located on the x-axis, 13 points from the origin.

I) Gives us the location of the midpoint of our segment

(45,65)

II) Gives us the location of one endpoint

(13,0)

Use I) and II) to work backwards with midpoint formula to find the other endpoint.

So endpoint is at .

Therefore, both statements are needed to answer the question.

To find the endpoint of a segment, we can generally use the midpoint formula; however, in this case we do not have enough information.

I) Gives us one endpoint

II) Gives us the length of DF

The problem is that we don't know the orientation of DF. It could go in infinitely many directions, so we can't find the location of without more information.

The midpoint formula is

,

with statement 1, we know that Q is and can solve for P:

and

Statement 1 alone is sufficient.

Statement 2 doesn't provide enough information to solve for point P.

You cannot determine the equation of a circle knowing only the center, as in Statement 1.

But given the coordinates of any diameter of a circle, as in Statement 2, you can use the midpoint formula to find the center, and the distance formula to find the distance from the center to either endpoint - this is the radius. Once you know the center and the radius, just apply them to the standard form of the equation of a circle.

The center of the circle of the equation is the point .

If Statement 2 is assumed, then ; since is known to be positive, is negative, which is the same as Statement 1.

From either statement, we know to be negative, and, subsequently, that both coordinates of the center are negative, putting it in Quadrant III.

The equation of a circle is given by:

Where r is the radius and (h,k) are the center of the circle

I) gives us (h,k)

By using II) and distance formula, we can find r

Thus, both statements are needed!

To write the equation of a circle we need the coordinates of its center and the length of its radius.

I) Gives us the center.

II) Tells us that the circle just touches the x-axis at the point (7,0), which is exactly 7 units below our center. This makes our radius 7.

Thus, we need both statements.

The equation of a circle is as follows:

Where is the center of the circle and is the radius.

Statement I tells us how to relate and , but doesn't tell us how to find either one.

Statement II gives us .

We almost have enough information here, but we can't tell for certain where the circle's center is, so more information is needed.

The equation of a circle is as follows:

Where is the center of the circle and is the radius.

Statement I gives us the radius.

Statement II gives us clues to find .

So, by using both, we can find the equation. Both are needed.

Recap:

Consider circle L.

I) The radius of L is twice the value of the third prime number.

II) The x-coordinate of L is four times the y-coordinate of L. The y-coordinate is equal to the radius.

Find the equation of circle L.

Using Statement I, the third prime number is 5, so our radius is 10.

Using Statement II, , so must be 40.

Putting it all into the equation of a circle:

The equation of a circle is as follows:

Where is the center of the circle and is the radius.

Statement I gives us a clue to find the diameter, which can be used to find the radius.

Statement II gives us the center of the circle.

Use Statement II and Statement I to find the radius, then plug it all into the equation to find the equation of Circle T.

Recap:

Circle T is plotted in Euclidean space. Find its equation.

I) The diameter of circle T is twice the square of the y-coordinate of its center.

II) Circle T is centered at the point .

Statement II gives us .

Use Statement I and Statement II to write the following equation:

So, if the diameter is 18, the radius must be 9. This makes our equation of the circle:

Circle equations are written in the form where are the coordinates for the center of the circle. When dealing with the area of a circle, the formula is Since, the circle equation already gives us the squared radius, we only need to multiply that number by Therefore, the area is

The slope of line is

From statement (1) we get another function of and . Therefore, we can calculate the values of and .

From

we can get .

Plug into , then we can get

and

Statement (2) only tells us the value of , which is useless to get the value of , because we have three unknown numbers with only two equations given.

Each statement alone gives one point of the parabola, which, even it is known to be the vertex, is not enough to determine its equation.

Assume both statements are true. The equation of a vertical parabola with vertex can be written as

Statement 1 gives the values of and as 2 and 4, respectively, so the equation of the parabola is

for some .

From Statement 2, we can substitute 4 and 7 for and , respectively, and solve

or

for , yielding the complete equation:

This makes the equation .

Statement 1 alone only gives one point of the parabola, which by itself does not determine its equation.

Statement 2 alone only gives two -intercepts, which are not sufficient to determine its equation; for example, the equations

and

are both equations of parabolas with their -intercepts at and .

Assume both statements are true. The standard form of the equation of a vertical parabola is

for some real , where is nonzero.

From each of the three given points, the - and -coordinates can be substituted in turn:

or

or

or

A system of three equations in three variables has been created:

Solving the three-by-three system yields the coefficients of the equation, so the two statements together provide sufficient information.

Assume both statements are true.

Statement 2 is actually a consequence of Statement 1 - the line of symmetry of a vertical parabola with two -intercepts is the line of the equation , where is the arithmetic mean of the -coordinates of the -intercepts. Therefore, we only need to examine Statement 1.

Statement 1 alone provides insufficient information, since we can demonstrate that at least two parabolas have the -intercepts given.

Parabola 1:

Substitute 0 for , and solve for :

or

and

Substitute 0 for , and solve for :

or

The equations are not equivalent, and both parabolas have -intercepts and .

Assume Statement 1 alone, which gives three points of the parabola.

The standard form of the equation of a vertical parabola is

for some real numbers , , and , where is non-zero.

From each of the three given points, the - and -coordinates can be substituted in turn:

or

or

or

A system of three equations in three variables has been created:

Solving the three-by-three system yields the coefficients of the equation, so Statement 1 alone provides sufficient information.

Statement 2 alone gives only the vertex and the line of symmetry, the latter of which is actually a consequence of the former; however, infinitely many parabolas have their vertices at , so Statement 2 alone provides insufficient information.

The equation of a parabola can be expressed in the form

,

for some nonzero , where is the vertex.

From Statement 1 alone, is the only -intercept; therefore, that is also the vertex. Substituting 4 and 0 for and , respectively, the equation becomes

,

or, simplified,

.

Since a third point, -intercept is given, we can substitute 0 and for and , respectively, and solve for :

The equation can be determined to be .

Now assume Statement 2 alone. We can examine these two equations:

Case 1:

We confirm that this parabola passes through using substitution:

Also, since this can be rewritten as

,

the vertex is , and the line of symmetry is the line with equation .

Case 2:

We confirm that this parabola passes through using substitution:

Also, since this can be rewritten as

,

the vertex is , and the line of symmetry is the line with equation .

Therefore, Statement 2 alone provides insufficient information.

The equation of a parabola can be expressed in the form

,

for some nonzero , where is the vertex.

From Statement 1, since is the only -intercept, it is the vertex. The equation is

,

or, simplified,

for some nonzero . But no further clues are given that could yield the value of .

Statement 2 alone only gives one point, which is not enough to determine the equation of a parabola.

Now assume both statements are true. Then, as stated before, the equation is

for some nonzero ; we can set up an equation by substituting 0 and 4 for and , respectively:

The equation is .

Statement 1 alone gives insufficient information; two points alone do not define a parabola. Statement 2 alone is insufficient; it only establishes that the -intercept is also the vertex.

Assume both statements are true. The equation of a parabola can be expressed in the form

,

for some nonzero , where is the vertex. Since we know two points that have the same -coordinate, 8, we can take the arithmetic mean of their -coordinates and find the -coordinate of the vertex:

From Statement 2, the parabola has exactly one -intercept, so that -intercept doubles as the vertex, and its -coordinate is 0.

We now know that the vertex is , and we know that, for some nonzero , the equation of the parabola is

,

or, simplified,

.

We can find by substituting 8 for both and :

The equation of the parabola has been determined to be .