Math Quiz: Take This Coordinate Geometry Questions!

Explanation
The mid-point of a line joining two points can be found by taking the average of their x-coordinates and the average of their y-coordinates. In this case, the x-coordinate of the mid-point is (a + 3a)/2 = 2a and the y-coordinate is (-b + 5b)/2 = 2b. Therefore, the mid-point is (2a, 2b).

Explanation
The point P which divides the line segment joining the points A(1,-3) and B(-3,9) internally in the ratio 1:3 is (0,0). This can be determined by using the section formula. The x-coordinate of P can be found by taking the weighted average of the x-coordinates of A and B, using the ratio 1:3. Similarly, the y-coordinate of P can be found by taking the weighted average of the y-coordinates of A and B, using the same ratio. Calculating these values, we get (0,0) as the coordinates of P.

Explanation
The line segment joining points A(3,4) and B(14,-3) can be represented by the equation of a line. By finding the equation of the line and substituting y=0 (since it meets the x-axis), we can solve for the x-coordinate of point P. The x-coordinate of point P is found to be 9.

To find the ratio in which P divides the segment AB, we can calculate the distance between A and P and the distance between P and B. The distance between A and P is 6, and the distance between P and B is 9. Therefore, the ratio in which P divides the segment AB is 6:9, which simplifies to 2:3.

Explanation
The centroid of a triangle is the point of intersection of its medians. The medians of a triangle are the line segments connecting each vertex to the midpoint of the opposite side. To find the centroid, we can find the midpoint of each side of the triangle and then find the point of intersection of the three midpoints. By calculating the midpoints of the sides of the triangle with vertices at (-2,-5), (-2,12), and (10,-1), we get midpoints at (4,3.5), (4.0,3.5), and (4,5.5). The point of intersection of these three midpoints is (4,4), which is the centroid of the triangle. Therefore, the given answer (2,2) is incorrect.

Explanation
In a parallelogram, opposite sides are equal in length. Therefore, the length of the side connecting (1,2) and (4,6) must be equal to the length of the side connecting (x,6) and (3,2). By calculating the distance between these two points using the distance formula, we can find that the length of the side connecting (1,2) and (4,6) is equal to the length of the side connecting (x,6) and (3,2). Solving for x, we find that x = 6.

Explanation
The area of a triangle can be calculated using the formula: A = 1/2 * base * height. In this case, the base of the triangle is the distance between the points (0,0) and (1,0), which is 1 unit. The height of the triangle is the distance between the points (0,1) and (0,0), which is also 1 unit. Therefore, the area of the triangle is 1/2 * 1 * 1 = 1/2 square units. Since none of the given options match this calculation, the correct answer is not available.

Explanation
The given points form a square with side length 1 unit. The area of a square is calculated by squaring the length of one of its sides. Therefore, the area of this square is 1 square unit.

Explanation
A straight line parallel to the X-axis has no vertical component, meaning it does not rise or fall. Therefore, it has an angle of inclination of 0 degrees.

Explanation
The slope of a line is determined by the difference in the y-coordinates divided by the difference in the x-coordinates between two points on the line. In this case, the slope is given as -3/2. To find the value of a, we can use the slope formula: (-2 - a) / (3 - (-1)) = -3/2. Simplifying this equation, we get (-2 - a) / 4 = -3/2. Cross-multiplying, we have -2 - a = -12/2. Simplifying further, we get -2 - a = -6. Solving for a, we find that a = 4.

Explanation
The slope of a line is determined by the change in y-coordinates divided by the change in x-coordinates. To find the slope of the line joining the points (-2,6) and (4,8), we use the formula: slope = (y2 - y1) / (x2 - x1). Plugging in the values, we get slope = (8 - 6) / (4 - (-2)) = 2 / 6 = 1/3. The slope of a line perpendicular to another line is the negative reciprocal of its slope. Therefore, the slope of the perpendicular line is -1/ (1/3) = -3.

Explanation
The correct answer is (1,7) because it satisfies both equations. By substituting x=1 and y=7 into the first equation, we get 9(1) - 7 - 2 = 0, which simplifies to 9 - 7 - 2 = 0, and 9 - 9 = 0. Similarly, substituting x=1 and y=7 into the second equation, we get 2(1) + 7 - 9 = 0, which simplifies to 2 + 7 - 9 = 0, and 9 - 9 = 0. Therefore, (1,7) is the point of intersection for both equations.

Explanation
The equation of the given straight line is 4x + 3y - 12 = 0. To find the point where it intersects the y-axis, we can set x = 0 and solve for y. When x = 0, the equation becomes 3y - 12 = 0. Solving for y, we get y = 4. Therefore, the point of intersection with the y-axis is (0, 4).

Explanation
The slope-intercept form of a straight line equation is y = mx + b, where m represents the slope. To find the slope of the given equation, we need to isolate y on one side of the equation. By rearranging the equation 7y - 2x = 11, we get y = (2/7)x + 11/7. Comparing this equation to the slope-intercept form, we can see that the slope is 2/7. Therefore, the correct answer is 2/7.

Explanation
The equation of a straight line parallel to the x-axis will have a constant y-value. Since the line passes through the point (2,-7), the y-value will be -7 for all x-values. Therefore, the equation of the line is y=-7.

Explanation
The x-intercept of a line is the point where the line crosses the x-axis, meaning that the y-coordinate is 0. The y-intercept is the point where the line crosses the y-axis, meaning that the x-coordinate is 0. To find the x-intercept, we set y=0 in the equation and solve for x. Similarly, to find the y-intercept, we set x=0 in the equation and solve for y. In this case, when we set y=0, we get 2x+6=0, which gives us x=-3. When we set x=0, we get -3y+6=0, which gives us y=2. Therefore, the x-intercept is -3 and the y-intercept is 2.

Explanation
The center of a circle is the midpoint of its diameter. In this case, the midpoint of the diameter is (-6,4). Since one end of the diameter is at (-12,8), the other end must be the same distance away from the center but in the opposite direction. Therefore, the other end of the diameter is at (0,0).

Explanation
The equation of a line passing through the origin and perpendicular to another line can be found by taking the negative reciprocal of the slope of the given line. The given line has a slope of -2/3, so the negative reciprocal is 3/2. Therefore, the equation of the line passing through the origin and perpendicular to the given line is 3x - 2y = 0.

Explanation
A straight line that is parallel to the y-axis will have a constant x-coordinate. Since the line passes through the point (-2,5), the x-coordinate is -2. Therefore, the equation of the line can be written as x = -2. Simplifying this equation gives x + 2 = 0.

Explanation
If the points (2,5), (4,6), and (a,a) are collinear, it means that they lie on the same straight line. To check if they are collinear, we can calculate the slope between the first two points and then compare it with the slope between the first and third points. The slope between (2,5) and (4,6) is (6-5)/(4-2) = 1/2. The slope between (2,5) and (a,a) is (a-5)/(a-2). Since these slopes should be equal, we can equate them: 1/2 = (a-5)/(a-2). Solving this equation, we find that a = 8. Therefore, the value of a is 8.

Explanation
Since the point (1,2) lies on the line y=2x+k, we can substitute the coordinates of the point into the equation to find the value of k. By substituting x=1 and y=2 into the equation, we get 2=2(1)+k, which simplifies to 2=2+k. Solving for k, we subtract 2 from both sides of the equation to get k=0. Therefore, the value of k is equal to 0.

Explanation
The equation of a straight line can be written in the form y = mx + c, where m is the slope and c is the y-intercept. In this case, the slope is given as 3 and the y-intercept is -4. Therefore, the equation of the line can be written as y = 3x - 4. By rearranging the equation, we get 3x - y - 4 = 0.

Explanation
The point of intersection of the straight lines y=0 and x=-4 is (-4,0) because the line y=0 represents the x-axis, and the line x=-4 represents a vertical line passing through the x-coordinate -4. Since the x-coordinate is fixed at -4, the y-coordinate can take any value. However, at the point of intersection, the y-coordinate is 0, indicating that the lines intersect at the point (-4,0).

Explanation
To determine the value of k for the given lines to be perpendicular, we need to find the slope of each line. The slope of the first line can be found by rearranging the equation into the slope-intercept form (y = mx + b), which gives us -1/2. The slope of the second line is found by rearranging the equation as well, resulting in k/2. For two lines to be perpendicular, the product of their slopes must be -1. Therefore, we can set up the equation (-1/2)(k/2) = -1 and solve for k. Simplifying, we get k = -4. Thus, the correct answer is -1.