Question 293 marksShort AnswerSection C

Find the ratio in which the x-axis divides the line segment joining the points (−6,5)(-6, 5) and (−4,−1)(-4, -1). Also, find the point of intersection.

Coordinate Geometry
Section Formula (Coordinate Geometry)

Official answer

The x-axis divides AB in the ratio 5:15:1, meeting it at (−133,0)\left(-\frac{13}{3}, 0\right). Let P divide A(−6,5)A(-6,5) and B(−4,−1)B(-4,-1) in ratio k:1k:1. Since P lies on the x-axis, its y-coordinate is 0: k(−1)+5k+1=0  ⟹  k=5\frac{k(-1)+5}{k+1} = 0 \implies k = 5. Substituting k = 5 in the x-coordinate formula: x=5(−4)+(−6)5+1=−266=−133x = \frac{5(-4)+(-6)}{5+1} = \frac{-26}{6} = \frac{-13}{3}. Hence the ratio is 5:15:1 and the point of intersection is (−133,0)\left(-\frac{13}{3}, 0\right).

Answer keywords:section formularatio of divisionx-axis intersectioncoordinate geometryinternal divisionpoint of intersection

Marking scheme

  1. 1.1 mark: correctly setting up the section formula for the y-coordinate: y=k(−1)+1(5)k+1y = \frac{k(-1) + 1(5)}{k+1} and equating to 0.
  2. 2.1 mark: correctly solving to get k=5k = 5, i.e., ratio 5:15:1.
  3. 3.1 mark: correctly substituting k=5k = 5 into the x-coordinate formula to get x=−133x = -\frac{13}{3}, giving point of intersection (−133,0)\left(-\frac{13}{3}, 0\right).

Hint

Let the ratio be k:1; use the section formula for the y-coordinate and set it to 0 (since the point lies on the x-axis) to solve for k, then find the x-coordinate.

Quick oral answer

I use the section formula for the y-coordinate, set it equal to zero since the point is on the x-axis, solve to get the ratio 5:15:1, then substitute back to find the x-coordinate, giving the point (−133,0)\left(-\frac{13}{3}, 0\right).

Common mistakes

  1. 1.Setting up the section formula with the points in the wrong order (swapping A and B), which flips the sign of the resulting ratio.
  2. 2.Forgetting that the point lies on the x-axis means y = 0, and instead trying to solve for both x and y simultaneously without using this condition first.
  3. 3.Arithmetic errors while simplifying −266-\frac{26}{6} to −133-\frac{13}{3}, or forgetting to state the final y-coordinate as 0 in the answer point.

Analysis and explanation

Set the y-coordinate of the section formula to zero (since the point lies on the x-axis) to find the ratio, then find the x-coordinate.


Concept

  • Section formula gives the coordinates of a point dividing AB in ratio k:1.
  • Any point on the x-axis has y-coordinate 0.

Key Points

  • Solve y=0y = 0 for k to get the ratio, then substitute k back to find x.
  • A above axis (y=5y=5) and B below (y=−1y=-1) confirms the segment does cross the x-axis internally.

Common Mistakes

  • Assuming the division is internal without checking the sign of the endpoints' y-coordinates.

Real-world/Exam Tip

  • The same technique (set x=0x = 0) finds where the y-axis divides a segment, and underlies centroid/area-of-triangle problems.

Interesting facts

The section formula used here is a direct extension of the midpoint formula — the midpoint formula is simply the special case of the section formula when the ratio is 1:11:1.

René Descartes introduced the coordinate system (Cartesian coordinates) in the 17th century, which is why coordinate geometry is also called 'Cartesian geometry' — this chapter is essentially the algebraic tool he pioneered for solving geometric problems.

In this problem, the fact that the y-coordinates of the two given points have opposite signs (+5 and −1) confirms geometrically that the segment must cross the x-axis somewhere between them, before any calculation is even done.

Question details

Q29 of 56 questions

Board
CBSE
Class
Class 10
Subject
Mathematics (Standard)
Year
2026
Section
Section C
Marks
3
Type
Short Answer
Chapter
Coordinate Geometry
Topic
Section Formula (Coordinate Geometry)
Bloom's level
apply
Syllabus topic
Unit VII: Coordinate Geometry — Chapter: Coordinate Geometry (Section formula; division of a line segment by the x-axis)
Author
CBSE
Reviewed by
Padhantu Academic Team

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Frequently asked questions

How do we know the division is internal and not external?

Because the ratio k came out positive (k=5k=5), and geometrically the y-coordinates of the two points (5 and −1) have opposite signs, confirming the x-axis passes between the two points, i.e., internal division.

What if the question instead asked about the y-axis dividing the segment?

The same method applies, except we would set the x-coordinate formula (instead of the y-coordinate formula) equal to 0 and solve for the ratio.