Exam-style CBSE practice questions on Section Formula (COORDINATE GEOMETRY). Try each one first, then reveal the correct answer and a step-by-step explanation. Free, from EduLevel — the AI teacher for CBSE.
The midpoint of the line segment connecting the points (a, 4) and (2, 2b) is (2, 6). Determine the value of a + b.
Recall the midpoint formula, which helps find the coordinates of the center point of a line segment given its endpoints.
Explanation: Using the midpoint formula, (a+2)/2 = 2 gives a=2, and (4+2b)/2 = 6 gives b=4. Therefore, a + b = 2 + 4 = 6.
Given two vertices of a triangle PQR as P(-1, 5) and Q(5, 2), find the coordinates of the point that divides the line segment PQ in the ratio 2:1.
To find a point that divides a line segment in a given ratio, you should recall the section formula.
Explanation: Applying the section formula with points P(-1, 5) and Q(5, 2) and ratio 2:1, the x-coordinate is (2*5 + 1*(-1))/(2+1) = 3 and the y-coordinate is (2*2 + 1*5)/(2+1) = 3. Thus, the coordinates are (3, 3).
On which of the following does the midpoint of the line segment connecting points P(-4, 5) and Q(4, 6) lie?
To find the midpoint of a line segment, you'll need to use the midpoint formula, which involves averaging the x-coordinates and the y-coordinates of the endpoints.
Explanation: The midpoint of the segment joining P(-4, 5) and Q(4, 6) is ((-4+4)/2, (5+6)/2) = (0, 5.5). Since the x-coordinate is 0, the point lies on the y-axis.
A line with the equation x/4 + y/6 = 1 intersects the x-axis at point P and the y-axis at point Q. What are the coordinates of the midpoint of the line segment PQ?
Remember how to find the points where a line intersects the x-axis and y-axis.
Explanation: The line intersects the x-axis at P(4, 0) (by setting y=0) and the y-axis at Q(0, 6) (by setting x=0). The midpoint of PQ is calculated as ((4+0)/2, (0+6)/2), which is (2, 3).
Consider the following statements: Assertion (A): The point that divides the line segment connecting A(1, 2) and B(-1, 1) internally in the ratio 1:2 has coordinates (-1/3, 5/3). Reason (R): The coordinates of a point dividing the line segment joining A(x₁, y₁) and B(x₂, y₂) in the ratio m₁:m₂ are given by the formula ((m₁x₂ + m₂x₁)/(m₁+m₂), (m₁y₂ + m₂y₁)/(m₁+m₂)). Which of the following is correct?
To evaluate the assertion, you need to use the section formula for internal division of a line segment.
Explanation: The Reason (R) states the correct section formula. However, applying this formula to the points in Assertion (A) yields the coordinates (1/3, 5/3), which contradicts the coordinates (-1/3, 5/3) given in the assertion. Therefore, Assertion (A) is false, while Reason (R) is true.
Find the ratio in which the x-axis divides the line segment connecting the points A(3, 6) and B(-12, -3).
Remember the section formula, which helps find the coordinates of a point dividing a line segment in a given ratio. Consider what special property the point dividing the line segment has when it lies on the x-axis.
Explanation: A point on the x-axis has a y-coordinate of 0. Using the section formula for the y-coordinate, we can set up the equation 0 = (k*y2 + 1*y1)/(k+1) to find the ratio k:1, which results in 2:1.
Given the positions of student A at (2,5) and student C at (8,6), what are the coordinates of the midpoint of the line segment AC?
To find the midpoint of a line segment, you need to consider the average of the x-coordinates and the average of the y-coordinates of the endpoints.
Explanation: The midpoint coordinates are calculated using the formula ((x1+x2)/2, (y1+y2)/2). For A(2,5) and C(8,6), this gives ((2+8)/2, (5+6)/2) = (5, 11/2).
Find the ratio in which the line segment connecting points A(-6, 10) and B(3, -8) is divided by the point P(-4, 6).
To find the ratio in which a line segment is divided by a point, you should recall the section formula.
Explanation: Let the ratio be k:1. Using the section formula for the x-coordinate, -4 = (k*3 + 1*(-6))/(k+1). This gives -4(k+1) = 3k - 6, which simplifies to -4k - 4 = 3k - 6. Solving this yields 7k = 2, so k = 2/7. The required ratio is 2:7.
If a point P divides the line segment joining student A(2,5) and student D(6,3) in the ratio 1:2, what are the coordinates of P?
To find the coordinates of a point that divides a line segment in a given ratio, you should recall the Section Formula.
Explanation: Using the section formula, P(x,y) = ((m1x2+m2x1)/(m1+m2), (m1y2+m2y1)/(m1+m2)), with ratio 1:2 for points A(2,5) and D(6,3), the coordinates are ((1*6+2*2)/(1+2), (1*3+2*5)/(1+2)) = (10/3, 13/3).
Determine the coordinates of the center of a circle, given that the endpoints of one of its diameters are (-6, 3) and (6, 4).
The center of a circle is the midpoint of any of its diameters. What is the formula for finding the midpoint of a line segment?
Explanation: The center of a circle is the midpoint of its diameter. Using the midpoint formula, the center is ((-6+6)/2, (3+4)/2) = (0/2, 7/2) = (0, 7/2).
The point P(k, 0) divides the line segment connecting A(2, -2) and B(-7, 4) in the ratio 1:2. What is the value of k?
Remember the section formula, which is used to find the coordinates of a point that divides a line segment in a given ratio.
Explanation: Using the section formula for the x-coordinate, k = (m*x2 + n*x1)/(m+n). Substituting the values, k = (1*(-7) + 2*2)/(1+2) = (-7+4)/3 = -3/3 = -1.
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