The point on the X-axis which is equidistant from the points (2, -5) and (-2, 9) is
1
(-7, 0)
2
(0, -7)
3
(7, 0)
4
(0, 7)
Official Solution
Correct Option: (1)
Step 1: Represent the point on the X-axis.
A point on the X-axis has coordinates of the form , since its -coordinate is .
Step 2: Use the distance formula.
The distance between two points and is given by:
Let the point on the X-axis be . The distances from to and must be equal. Thus:
Step 3: Simplify the equation.
Simplify both sides:
Square both sides to eliminate the square roots:
Expand both sides:
Simplify:
Cancel from both sides:
Rearrange terms:
Step 4: Write the coordinates of the point.
The point on the X-axis is .
Final Answer: The point on the X-axis is , which corresponds to option .
02
PYQ 2020
medium
mathematicsID: ap-polyc
The ratio in which the X-axis divide's the line segment joining the points (4, 6) and (3.-8) is
1
1:2
2
2:3
3
3:4
4
4:5
Official Solution
Correct Option: (3)
Step 1: Recall the section formula.
If a point divides a line segment joining two points and in the ratio , then the coordinates of the dividing point are given by:
Here, the dividing point lies on the X-axis, so its -coordinate is .
Step 2: Use the condition for the X-axis.
The -coordinate of the dividing point is . Using the section formula for the -coordinate:
Substitute and :
Simplify:
Rearrange:
Step 3: Interpret the result.
The ratio is .
Final Answer: The ratio in which the X-axis divides the line segment is , which corresponds to option .
03
PYQ 2020
medium
mathematicsID: ap-polyc
The mid-point of the line joining the points (4, 5) and (-2, -1) is
1
(1, 3)
2
(3, 1)
3
(1, 2)
4
(2, 1)
Official Solution
Correct Option: (3)
Step 1: Recall the formula for the midpoint.
The midpoint of a line segment joining two points and is given by:
Step 2: Substitute the coordinates of the given points.
The given points are and . Substituting into the formula:
Step 3: Simplify the calculations.
Final Answer: The midpoint is , which corresponds to option .
04
PYQ 2022
medium
mathematicsID: ap-polyc
In the given figure, O is the centre of the circle and ZAOC = 110°, then ZADC is equal to
1
110°
2
55°
3
70°
4
125°
Official Solution
Correct Option: (2)
We are given that is the centre of the circle and . In the case of a circle, the angle at the centre of the circle is twice the angle subtended at the circumference by the same chord. This means: Substituting the given value of : Now, solving for :
The correct option is (B):
05
PYQ 2022
medium
mathematicsID: ap-polyc
If the points P(2, 3), Q(5, k) and R(6, 7) are collinear, then the value of k is
1
4
2
3
4
6
Official Solution
Correct Option: (4)
We are given that the points , , and are collinear. For the points to be collinear, the slope between any two points must be the same. Let's calculate the slope between points and , and the slope between points and . The slope between two points and is given by the formula: 1. Slope between and : 2. Slope between and : Since the points are collinear, the slopes must be equal: Now, solve for :
The correct option is (D):
06
PYQ 2024
easy
mathematicsID: ap-polyc
In the following figure, if DE || BC, then x =
1
2
3
4
Official Solution
Correct Option: (2)
Since DE is parallel to BC, the BPT states that the ratio of corresponding sides is equal.
Therefore: AD/AB = AE/AC = DE/BC
Substitute the given values: (x + 4) / (x + 4 + x + 3) = (2x - 1) / (2x - 1 + x + 1)
Simplify: (x + 4) / (2x + 7) = (2x - 1) / (3x)
Cross-multiply: 3x(x + 4) = (2x + 7)(2x - 1)
Expand: 3x² + 12x = 4x² + 12x - 7
Solve for x:** 0 = x² - 7 x² = 7 x = √7
So, the correct answer is (2) √7.
07
PYQ 2025
medium
mathematicsID: ap-polyc
and are the midpoints of sides and of a triangle respectively and cm. If , then the length of is
1
cm
2
cm
3
cm
4
cm
Official Solution
Correct Option: (2)
Step 1: Recall the Midpoint Theorem. The Midpoint Theorem states that the line segment connecting the midpoints of two sides of a triangle is parallel to the third side and is half the length of the third side. Step 2: Apply the Midpoint Theorem to the given triangle. In triangle , is the midpoint of side , and is the midpoint of side . According to the Midpoint Theorem, the line segment is parallel to the third side , and its length is half the length of . Step 3: Calculate the length of . Given that cm, the length of is: The length of is 5 cm.
08
PYQ 2025
medium
mathematicsID: ap-polyc
If is the midpoint of the line segment joining and , then
1
2
3
4
Official Solution
Correct Option: (4)
Step 1: Recall the midpoint formula. The midpoint of a line segment joining two points and is given by: Step 2: Apply the midpoint formula to the given points. We are given , , and the midpoint . Using the midpoint formula for the x-coordinate: Using the midpoint formula for the y-coordinate: Step 3: Solve for using the x-coordinate equation. Multiply both sides by 3 to find : Step 4: Verify the y-coordinate. Let's check if the y-coordinate of the midpoint matches: The y-coordinate matches the given midpoint. Thus, the value of is .
