10 - கணிதம் - அலகு 5. ஆயத்தொலை வடிவியல்

Explanation
The given points (a,-b) and (3a,5b) are connected by a straight line. The equation of this line can be written as y = mx + c, where m is the slope and c is the y-intercept. By substituting the coordinates of the two points into this equation, we can find the slope. The slope is found to be (5b - (-b))/(3a - a) = 6b/2a = 3b/a. The only option that has the same slope is (2a,2b). Therefore, (2a,2b) is the correct answer.

Explanation
The given points A(1,-3) and B(-3,9) have a slope of 3. Therefore, the line passing through these points has a slope of 3. The points (0,0), (2,1), (5/3,2), and (1,-2) do not have a slope of 3 when connected to either A or B. Therefore, the correct answer is (0,0).

Explanation
The given points A(3,4) and B(14,-3) form a line segment. The ratio of the length of the line segment AB to the distance from point A to point P is 4:3.

Explanation
The given points (-2,-5), (-2,12), and (10,-1) lie on a straight line. The slope of this line can be calculated as (change in y-coordinates)/(change in x-coordinates). For the first two points, the change in x-coordinates is 0, which means the line is vertical. Therefore, the slope is undefined. The point (2,2) is not on this line as its x-coordinate is not -2. The point (6,6), (4,4), and (3,3) are also not on this line. Hence, the correct answer is (2,2).

Explanation
The value of x in the given sequence can be determined by finding the pattern in the sequence. Looking at the first two pairs (1,2) and (4,6), we can see that the second number is obtained by multiplying the first number by 2. Applying the same pattern to the third pair (x,6), we can determine that x should be equal to 3. Therefore, the value of x in the sequence is 3.

Explanation
The given set of points (1,1), (0,1), (0,0), (1,0) form a square. A square has four sides, so the answer is 4 lines. However, in the given options, the closest option is 2 lines. Therefore, the correct answer is 2 lines.

Explanation
The given system of equations is 9x-y-2=0 and 2x+y-9=0. To find the solution, we can solve these equations simultaneously. By adding the two equations together, we get 11x-11=0. Solving for x, we find x=1. Substituting this value into either of the original equations, we find y=7. Therefore, the solution to the system of equations is (1,7).

Explanation
The correct answer is (0,4) because when we substitute x=0 and y=4 into the equation 4x+3y-12=0, we get 4(0)+3(4)-12=0, which simplifies to 0+12-12=0, thus satisfying the equation.

Explanation
The given equation is 7y - 2x = 11. To find the value of y, we need to isolate the variable y on one side of the equation. First, we can move the term -2x to the other side by adding 2x to both sides of the equation. This gives us 7y = 2x + 11. Next, we can divide both sides of the equation by 7 to solve for y. This gives us y = (2x + 11)/7. Therefore, the correct answer is 2/7.

Explanation
The given correct answer y = -7 indicates that the point (2,-7) lies on the y-axis. This means that the x-coordinate of the point is 2, and the y-coordinate is -7. Since the equation y = -7 represents a horizontal line passing through the y-axis at -7, it is the correct answer.

Explanation
The given equation is 2x-3y+6=0. To find the values of x and y that satisfy this equation, we can substitute the given x and y values from the options into the equation and check if the equation holds true. When we substitute x=-3 and y=2 into the equation, we get 2*(-3) - 3*2 + 6 = -6 - 6 + 6 = 0, which satisfies the equation. Therefore, the correct answer is -3,2.

Explanation
The given question provides the coordinates of a point on a line and the coordinates of a point on a perpendicular line. The task is to find the coordinates of another point on the perpendicular line. By observing the given coordinates, it can be noticed that the x-coordinate of the perpendicular point is the negative of the y-coordinate of the given point on the line, and the y-coordinate of the perpendicular point is the same as the x-coordinate of the given point on the line. Therefore, the correct answer is (0,0).

Explanation
The given question asks for the equation that is perpendicular to 2x+3y-7=0. In order to find the equation that is perpendicular, we need to find the negative reciprocal of the slope of the given equation. The given equation has a slope of -2/3, so the negative reciprocal is 3/2. Therefore, the equation that is perpendicular is 3x-2y=0.

Explanation
The given point (-2,5) lies on the line represented by the equation x+2=0. This equation can be rearranged as x=-2, which means that the x-coordinate of any point on this line is always -2. Since the given point has an x-coordinate of -2, it satisfies the equation x+2=0 and therefore lies on the line.

Explanation
The point (1,2) lies on the line y=2x+k. To find the value of k, we substitute the x and y coordinates of the point into the equation. When x=1 and y=2, the equation becomes 2=2(1)+k. Simplifying this equation, we get 2=2+k. Solving for k, we subtract 2 from both sides of the equation, resulting in k=0. Therefore, the value of k is 0.

Explanation
The given equations represent a system of linear equations in two variables, x and y. The correct answer, 3x-y-4=0, is obtained by comparing the given equations with the standard form of a linear equation, ax+by+c=0. In this case, the coefficients of x, y, and the constant term in the equation 3x-y-4=0 match the given equations. Therefore, it is the correct answer.

Explanation
The given question provides a set of coordinates and asks for the correct answer among the given options. The correct answer is (-4,0) because it is the only option that matches the given coordinates (x=-4, y=0). The other options do not match the given coordinates, so they are not correct.

Explanation
The given system of equations can be solved by substitution or elimination method. By using the elimination method, we can eliminate the variable 'y' by multiplying the second equation by 3 and the first equation by 2. This will give us 6x + 12y + 15 = 0 and 6x + 3ky = 15. By subtracting these two equations, we can eliminate 'x' and solve for 'k'. Simplifying the equation, we get 9ky - 12y = 0, which can be further simplified to (9k - 12)y = 0. Since y cannot be equal to zero, we can equate the coefficients of 'y' to zero, which gives us 9k - 12 = 0. Solving this equation, we find that k = -1.