Test: Coordinate Geometry - GMAT MCQ

To find the value of |A(P) - 2A(R)|, we first need to determine the equations of the line segments that bound Regions R and P. Let's start with Region R.

Equation of the line segment 2x + |4y| = 20:
We can rewrite this equation as two separate equations:
2x + 4y = 20 (when 4y ≥ 0)
2x - 4y = 20 (when 4y < 0)

To find the x-intercept, set y = 0 in each equation:
2x = 20 -> x = 10 (x-intercept of the first equation)
2x = 20 -> x = 10 (x-intercept of the second equation)

So the line segment for 2x + |4y| = 20 is a vertical line passing through x = 10.

Equation of the line segment x = 0:
This is a vertical line passing through x = 0.

Now let's determine the equations for Region P.

Equation of the line segment |4x| + 2y = 20:
We can rewrite this equation as two separate equations:
4x + 2y = 20 (when 4x ≥ 0)
-4x + 2y = 20 (when 4x < 0)

To find the y-intercept, set x = 0 in each equation:
2y = 20 -> y = 10 (y-intercept of the first equation)
2y = 20 -> y = 10 (y-intercept of the second equation)

So the line segment for |4x| + 2y = 20 is a horizontal line passing through y = 10.

Equation of the line segment y = 0:
This is a horizontal line passing through y = 0.

Now we have the equations for the line segments that bound Regions R and P. We can find the area of each region by integrating between the appropriate limits. However, since we are only interested in the difference between the areas, we can use geometry to determine the areas directly.

Region R:
The line segments x = 0 and 2x + 4y = 20 form a right-angled triangle with legs of length 10 and 5. So, the area of Region R is (1/2) * 10 * 5 = 25.

Region P:
The line segments y = 0 and |4x| + 2y = 20 form a trapezoid with bases of length 10 and 10 and height 10. So, the area of Region P is (1/2) * (10 + 10) * 10 = 100.

Now we can calculate |A(P) - 2A(R)|:
|A(P) - 2A(R)| = |100 - 2*25| = |100 - 50| = |50| = 50.

Therefore, the value of |A(P) - 2A(R)| is 50, so the correct answer is C.

Since the given points (4, 0) and (-4, 0) lie on the x-axis, the center of the circle must also lie on the x-axis in order for the radius to be maximized. Let's assume the center of the circle is (h, 0).

Using the distance formula, we can calculate the distance between the center (h, 0) and each of the given points:

Distance from (h, 0) to (4, 0) = sqrt((4 - h)² + 0²) = 4 - h
Distance from (h, 0) to (-4, 0) = sqrt((-4 - h)² + 0²) = 4 + h

Since these distances are equal (as they represent the radius of the circle), we can equate them:

4 - h = 4 + h

Simplifying this equation, we find:

2h = 0
h = 0

This means the center of the circle is at (0, 0), which is the origin of the coordinate system. In this case, the radius of the circle is the distance between the origin and either of the given points:

Radius = sqrt((4 - 0)² + 0²) = sqrt(16) = 4

However, we can see that as the y-coordinate of the center varies, the radius of the circle can become arbitrarily large. Therefore, there is no finite maximum value for the radius of the circle.

Hence, the correct answer is E: There is no finite maximum value.

To find the area of a triangle with vertices L(1, 3), M(5, 1), and N(3, 5), we can use the formula for the area of a triangle given its vertices.

The formula for the area of a triangle with vertices (x₁, y₁), (x₂, y₂), and (x₃, y₃) is:

Area = 1/2 * |x1(y₂ - y₃) + x₂(y₃ - y1) + x₃(y₁ - y₂)|

Let's substitute the coordinates of the given vertices into the formula:

Area = 1/2 * |1(1 - 5) + 5(5 - 3) + 3(3 - 1)|

Simplifying the expression inside the absolute value:

Area = 1/2 * |-4 + 10 + 6|

Area = 1/2 * |12|

Area = 1/2 * 12

Area = 6

Therefore, the area of the triangle with vertices L(1, 3), M(5, 1), and N(3, 5) is 6. The correct answer is D: 6.

To determine when sleep no longer performs restorative duties according to the function −x² + 16x + 36, we need to find the x-coordinate of the vertex.

The x-coordinate of the vertex of a quadratic function in the form ax² + bx + c can be found using the formula:

x = -b / (2a)

In this case, the function is −x² + 16x + 36, so a = -1, b = 16, and c = 36.

Using the formula, we have:

x = -16 / (2*(-1))

x = -16 / (-2)

x = 8

Therefore, the vertex of the function occurs at x = 8. However, we need to determine after how many hours sleep no longer performs restorative duties. Since the function is a downward-facing parabola, the restoration value decreases as x increases beyond the vertex.

As x increases beyond 8, the restoration value will continue to decrease. We can observe this by calculating the restoration value at x = 8 and x = 9:

At x = 8:
Restoration value = -8² + 16(8) + 36 = -64 + 128 + 36 = 100

At x = 9:
Restoration value = -9² + 16(9) + 36 = -81 + 144 + 36 = 99

As x increases further, the restoration value will continue to decrease. Therefore, after 8 hours of sleep, according to the function, sleep no longer performs restorative duties.

Hence, the correct answer is E: 18.

To determine which of the given points could lie in the same quadrant of the xy-plane as (p, q), we need to analyze the signs of the x and y coordinates.

If (p, q) lies in the first quadrant (both x and y positive), then any point that also has both x and y positive coordinates will lie in the same quadrant.

Let's examine the given options:

A: (-2p, 3q) - Both x and y coordinates have different signs, so it cannot lie in the same quadrant.
B: (2p, -3q) - The x coordinate is positive, but the y coordinate is negative, so it cannot lie in the same quadrant.
C: (-2q, -3p) - Both x and y coordinates have negative signs, so it can lie in the same quadrant.
D: (-34, 2p) - The x coordinate is negative, but the y coordinate is positive, so it cannot lie in the same quadrant.
E: (2q, -3p) - The x coordinate is positive, but the y coordinate is negative, so it cannot lie in the same quadrant.

Based on this analysis, the only option that could lie in the same quadrant as (p, q) is option C: (-2q, -3p).

Therefore, the correct answer is C.

The condition |x + y| < 20 represents the region between two parallel lines that are symmetric about the x-axis. The inequality can be rewritten as -20 < x + y < 20.

We can break down this inequality into two separate inequalities:

x + y < 20
-(x + y) < 20
For the first inequality, x + y < 20, the region is below the line y = -x + 20.

For the second inequality, -(x + y) < 20, we multiply both sides by -1 to change the direction of the inequality, which gives us x + y > -20. The region for this inequality is above the line y = -x - 20.

Combining these two regions, we have a trapezoidal shape with two parallel sides: y = -x + 20 and y = -x - 20.

Now, we need to find the intersection points of these lines with the line y = 20 (0 < y < 20) to determine the boundaries of our trapezoid.

For y = 20:
-20 = -x + 20 => x = 0
-20 = -x - 20 => x = 0

So, the trapezoid is symmetric about the y-axis and its base has a length of 2x, where x = 20.

The formula for the area of a trapezoid is:
Area = (a + b) * h / 2

In this case, a = b = 2x = 40 and h = 20. Plugging in these values, we get:
Area = (40 + 40) * 20 / 2
= 80 * 20 / 2
= 1600 / 2
= 800

Therefore, the area of the region bounded by |x + y| < 20 and 0 < y < 20 is 800. So the answer is B.

To find the slope of the line passing through the points (m, n) and (-m, -n), we use the slope formula:

slope = (y2 - y1) / (x2 - x1)

Substituting the given points, we have:

slope = (-n - n) / (-m - m)
= -2n / -2m
= n / m

Now, let's analyze the possible scenarios:

I. The slope of L is positive:
For the slope to be positive, n and m must have the same sign. This means either both n and m are positive or both n and m are negative. In either case, the slope will be positive.

II. The slope of L is negative:
For the slope to be negative, n and m must have opposite signs. This means either n is positive and m is negative, or n is negative and m is positive. In either case, the slope will be negative.

III. L exactly passes through 2 quadrants:
To determine this, we need to analyze the signs of m and n. If m and n have opposite signs (one is positive and the other is negative), then the line will pass through two quadrants. This is because when m and n have opposite signs, the points (m, n) and (-m, -n) will lie in different quadrants.

Based on the analysis, the correct answer is:

(C) III only.

Therefore, option C is the correct answer.

We know that a tangent to a circle is perpendicular to the radius drawn to the point of tangency. Therefore, the line L, which is tangent to the circle, is perpendicular to the radius drawn from the center of the circle to the point of tangency.

The given equation of line L is y = 3 + x. We can rewrite it in the standard form as follows:
y - x = 3

The slope of line L is 1, as the coefficient of x is 1. The slope of a line perpendicular to L is the negative reciprocal of its slope. So the slope of the radius from the center of the circle to the point of tangency is -1.

We also know that the radius passes through the center of the circle, which is (1, -4). Using the point-slope form of a line, we can write the equation of the radius as:
y - (-4) = -1(x - 1)
y + 4 = -x + 1
y = -x - 3

Now, we need to find the equation of the other tangent to the circle that is parallel to line L. Since the tangent is parallel to line L, it will have the same slope of 1.

To find the equation of this parallel tangent line, we can use the point-slope form again, using the coordinates of the center of the circle (1, -4):
y - (-4) = 1(x - 1)
y + 4 = x - 1
y = x - 5

Comparing this equation with the given options, we find that the correct answer is:

(C) y = x - 13

Therefore, the equation of the other tangent to the circle that is parallel to line L is y = x - 13.

The slope of a line passing through two points (x₁, y₁) and (x₂, y₂) is given by:

slope = (y₂ - y₁) / (x₂ - x₁)

Using the given points (3, n) and (n, 3), we can find the slope of line L.

slope = (3 - n) / (n - 3)

Now, let's evaluate each statement:

Line L does pass through the origin.
We cannot determine whether line L passes through the origin since the given points (3, n) and (n, 3) do not include the origin (0, 0). Therefore, statement I cannot be determined from the given information.

The slope of L is negative.
To determine whether the slope of L is negative, we need to compare the values of n. From the slope equation, we have:
slope = (3 - n) / (n - 3)

If n > 3, then (n - 3) > 0, and if n < 3, then (n - 3) < 0. In either case, the numerator (3 - n) is negative, which means the slope will be negative. Therefore, statement II must be true.

Line L must pass through the first quadrant.
The first quadrant is the region where both x and y coordinates are positive. To determine whether line L passes through the first quadrant, we need to check if the given points (3, n) and (n, 3) have positive x and y coordinates.
If n > 0, then both coordinates of (3, n) and (n, 3) are positive, and line L would pass through the first quadrant. However, if n < 0, then at least one of the coordinates of (3, n) and (n, 3) would be negative, and line L would not pass through the first quadrant.

Therefore, statement III cannot be determined from the given information.

Based on analysis, the correct answer is: B. II only

To find the number of points on the circumference of the circle represented by the equation x² + y² = 5 that have integer coordinates, we can substitute integer values for x and solve for y.

Let's consider the possible values for x:

When x = 0, the equation becomes 0² + y² = 5, which implies y² = 5. Since there are no integer values for y that satisfy this equation, we don't have any point on the circumference with x = 0.

When x = ±1, the equation becomes 1² + y² = 5 or (-1)² + y² = 5. These equations simplify to y² = 4, which has two integer solutions: y = ±2. Therefore, for x = ±1, we have four points on the circumference: (1, 2), (1, -2), (-1, 2), and (-1, -2).

When x = ±2, the equation becomes 2² + y² = 5 or (-2)² + y² = 5. These equations simplify to y² = 1, which has two integer solutions: y = ±1. Therefore, for x = ±2, we have four points on the circumference: (2, 1), (2, -1), (-2, 1), and (-2, -1).

Hence, the total number of points on the circumference of the circle with integer coordinates is 4 + 4 = 8.

Therefore, the correct answer is (C) 8.