Test: Coordinate Geometry - GMAT MCQ

To find the coordinates of the center of the circle that circumscribes triangle ABC, we can use the properties of the circumcenter.

The circumcenter is the intersection point of the perpendicular bisectors of the sides of the triangle. These perpendicular bisectors are the lines that pass through the midpoints of each side and are perpendicular to those sides.

Let's calculate the midpoints of the sides AB, BC, and AC:

Midpoint of AB = ((1+4)/2, (2+2)/2) = (2.5, 2) Midpoint of BC = ((4+4)/2, (2+6)/2) = (4, 4) Midpoint of AC = ((1+4)/2, (2+6)/2) = (2.5, 4)

Next, we need to find the equations of the perpendicular bisectors of AB and BC.

The slope of AB is (2-2)/(4-1) = 0/3 = 0. The negative reciprocal of 0 is undefined. Therefore, the equation of the perpendicular bisector of AB is a vertical line passing through the midpoint (2.5, 2), which can be written as x = 2.5.

The slope of BC is (6-2)/(4-4) = 4/0, which is undefined. Therefore, the equation of the perpendicular bisector of BC is a horizontal line passing through the midpoint (4, 4), which can be written as y = 4.

The point of intersection of these two lines, x = 2.5 and y = 4, gives us the coordinates of the circumcenter.

Hence, the coordinates of the center of the circle that circumscribes triangle ABC are (2.5, 4).

Therefore, the correct answer is B: (2.5, 4).

Let the first term be x₁ and the last term be x₁₀ 
0.8x₁ − 24____________0.8x₁₀ − 24 
We know that
0.8x₁₀−24 − (0.8x₁ − 24) = 7 
0.8x₁₀ − 24−0.8x₁+24 = 7
0.8(x₁₀−x₁) = 7
x₁₀ − x₁ = 7∗10/8 
x₁₀ − x₁ = 8.75 

To determine which of the given points must lie on the line mx + 4y = 8 for every possible value of m, we can substitute the coordinates of each point into the equation and see which points satisfy it for all values of m.

Let's check each point:

I. (2, 0) Substituting x = 2 and y = 0 into the equation: m(2) + 4(0) = 8 2m = 8 m = 4

Since m = 4 satisfies the equation, point (2, 0) lies on the line for this particular value of m. However, we need to consider all possible values of m, so this point does not satisfy the equation for every value of m.

II. (2, 2) Substituting x = 2 and y = 2 into the equation: m(2) + 4(2) = 8 2m + 8 = 8 2m = 0 m = 0

Since m = 0 satisfies the equation, point (2, 2) lies on the line for this particular value of m. However, we need to consider all possible values of m, so this point does not satisfy the equation for every value of m.

III. (0, 2) Substituting x = 0 and y = 2 into the equation: m(0) + 4(2) = 8 8 = 8

Since the equation is true regardless of the value of m, point (0, 2) lies on the line for every possible value of m.

Based on our analysis, only point III, (0, 2), must lie on the line mx + 4y = 8 for every possible value of m.

Therefore, the correct answer is C: III only.

To find the coordinates of the point that divides the line joining (5, -2) and (9, 6) internally in the ratio 1:3, we can use the section formula.

The section formula states that if a line segment AB is divided by a point P in the ratio m:n, the coordinates of point P can be found using the following formulas:

Px = (n * Ax + m * Bx) / (m + n) Py = (n * Ay + m * By) / (m + n)

In this case, we want to find the coordinates of the point that divides the line joining (5, -2) and (9, 6) internally in the ratio 1:3. Therefore, m = 1 and n = 3.

Using the section formula, we can calculate the coordinates of the point:

Px = (3 * 5 + 1 * 9) / (1 + 3) = (15 + 9) / 4 = 24 / 4 = 6 Py = (3 * -2 + 1 * 6) / (1 + 3) = (-6 + 6) / 4 = 0 / 4 = 0

Therefore, the coordinates of the point that divides the line internally in the ratio 1:3 are (6, 0).

Hence, the correct answer is A: (6, 0).

To determine how many points a circle can intersect the coordinate axes, we need to consider the possible scenarios.

1. The circle can intersect an axis at a single point: This occurs when the center of the circle lies on the axis, and the radius is equal to the distance from the center to the axis.

2. The circle can intersect an axis at two points: This occurs when the center of the circle is located above or below the axis, and the radius is greater than the distance from the center to the axis.

3. The circle can intersect both axes: This occurs when the center of the circle is not on either axis, and the radius is greater than the distance from the center to both axes.

4. The circle may not intersect any axis: This occurs when the center of the circle is located outside the coordinate plane or when the radius is smaller than the distance from the center to the axes.

Based on these scenarios, it is clear that a circle can intersect the coordinate axes at a minimum of 1 point and a maximum of 4 points. Therefore, the correct answer is D: 1, 2, 3, or 4 only.

If the segment X, represented by -1 < x < 5, is shifted 5 units in the positive direction, the new range of X becomes 4 < x < 10.

Similarly, if the segment Y, represented by 6 ≤ y ≤ 10, is shifted 2 units in the positive direction, the new range of Y becomes 8 ≤ y ≤ 12.

By comparing the new ranges, we can determine the common integers between X and Y, which are 8 and 9.

Therefore, the answer is B: 2 common integers.

To find the value of k, we can set up an equation using the given information.

The equation of the line 2x + 3y = k can be rewritten as 3y = -2x + k and further simplified to y = (-2/3)x + (k/3).

The region enclosed by the line and the two axes forms a right triangle. The area of a triangle is given by the formula A = (base * height) / 2.

In this case, the base of the triangle is the x-intercept, which occurs when y = 0. Setting y = 0 in the equation y = (-2/3)x + (k/3), we get:

0 = (-2/3)x + (k/3) 2x = k

So, the x-intercept is x = k/2.

The height of the triangle is the y-intercept, which occurs when x = 0. Setting x = 0 in the equation y = (-2/3)x + (k/3), we get:

y = k/3

So, the y-intercept is y = k/3.

The area of the triangle is given as 6 square units. Therefore, we can write the area equation as:

(1/2) * (base) * (height) = 6 (1/2) * [(k/2) * (k/3)] = 6 (k² / 12) = 6 k² = 72 k = √72

Simplifying further, we have k = √(36 * 2) = 6√2.

Therefore, the value of k is 6√2.

Hence, the correct answer is a: k = 6√2.

Since the line 'p' passes through the origin and has a slope of 3, its equation can be written as y = 3x.

Given that the points (x, 12) and (6, y) lie on the line 'p', we can substitute the coordinates into the equation to find their respective y-values:

For the point (x, 12): 12 = 3x x = 4

For the point (6, y): y = 3(6) y = 18

Now, we can calculate the value of x - 2y: x - 2y = 4 - 2(18) = 4 - 36 = -32

Therefore, the value of x - 2y is -32.

Hence, the correct answer is c: -32.

The line L has a slope of 2 and a y-intercept of 4. This means the equation of the line can be written as y = 2x + 4.

We are given that the point (1, y) lies on the line. Substituting x = 1 into the equation, we get y = 2(1) + 4 = 6.

We are also given that the point (x, 2) lies on the line. Substituting y = 2 into the equation, we get 2 = 2x + 4. Solving this equation for x, we subtract 4 from both sides: 2 - 4 = 2x, resulting in -2 = 2x. Dividing both sides by 2, we find x = -1.

Now, we can calculate x - y:

x - y = -1 - 6 = -7.

Therefore, the value of x - y is -7.

Hence, the correct answer is B: -7.

To find the perpendicular distance between two parallel lines, we can use the formula:

Perpendicular distance = |b2 - b1| / √(m1^2 + 1)

where (m1, b1) and (m2, b2) are the slopes and y-intercepts of the two lines, respectively.

In this case, the two parallel lines are y = 4x + 5 and y = 4x + 7.

Comparing the equations, we can see that the slopes (m1 and m2) are both equal to 4, and the y-intercepts (b1 and b2) are 5 and 7, respectively.

Using the formula, we can calculate the perpendicular distance:

Perpendicular distance = |7 - 5| / √(4^2 + 1) Perpendicular distance = 2 / √(16 + 1) Perpendicular distance = 2 / √17

Simplifying further:

Perpendicular distance = 2 / √17 * (√17 / √17) Perpendicular distance = 2√17 / 17

Therefore, the perpendicular distance between the two parallel lines is 2/√17 units.

Hence, the correct answer is B: 2/√17 units.