Question 293 marksShort AnswerSection C

  1. 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 - ratio of division by x-axis

Official answer

The x-axis divides the segment joining A(−6,5)A(-6, 5) and B(−4,−1)B(-4, -1) in the ratio 5:15 : 1, and the point of intersection is (−133,0)(-\frac{13}{3}, 0). The ratio is internal (k=5>0)(k = 5 > 0) since A lies above the x-axis and B lies below it.

Answer keywords:section formulax-axis dividesratio k:1y-coordinate zeroratio 5:1point of intersection(−13/3, 0)internal division

Marking scheme

  1. 1.1 mark: assuming ratio k:1k : 1 and correctly applying the section formula for the y-coordinate: (−k+5)/(k+1)=0(-k + 5)/(k + 1) = 0.
  2. 2.1 mark: solving to get k=5k = 5, i.e. the ratio 5:15 : 1.
  3. 3.1 mark: substituting into the x-coordinate formula to obtain x=−133x = -\frac{13}{3} and stating the point as (−133,0)(-\frac{13}{3}, 0).

Hint

Take the ratio as k:1k : 1 and use the y-coordinate of the section formula = 0 (because points on the x-axis have y=0y = 0) to find k, then substitute k into the x-formula.

Quick oral answer

I take the ratio as k:1k : 1; since the crossing point is on the x-axis its y-coordinate is 0, so (−k+5)/(k+1)=0(-k + 5)/(k + 1) = 0 gives k=5k = 5, i.e. 5:15 : 1, and putting k=5k = 5 in the x-formula gives x=−133x = -\frac{13}{3}, so the point is (−133,0)(-\frac{13}{3}, 0).

Common mistakes

  1. 1.Using the x-coordinate condition to find the ratio instead of the y=0y = 0 condition — points on the x-axis are defined by y=0y = 0, not x=0x = 0.
  2. 2.Interchanging (x1,y1)(x_1, y_1) and (x2,y2)(x_2, y_2), which flips the ratio (e.g. getting 1:51 : 5 instead of 5:15 : 1) and gives a wrong point.
  3. 3.Sign errors with the negative coordinates when computing (5(−4)+(−6))/6(5(-4) + (-6))/6, leading to a wrong x-value such as +133+\frac{13}{3} or −265-\frac{26}{5}.

Analysis and explanation

Use the section formula with the x-axis condition y=0y = 0 to find both the dividing ratio and the intersection point.


Concept

  • Any point on the x-axis has y-coordinate 0, so assume the ratio as k:1k : 1 (a single unknown) and set the section-formula y-coordinate to 0.
  • Solving gives k=5k = 5, so the ratio is 5:15 : 1 — since k>0k > 0, the division is internal, confirming the segment genuinely crosses the axis (A is above it, B below it).
  • Reusing the same k in the x-coordinate formula gives the intersection point (−133,0)(-\frac{13}{3}, 0).

Common mistakes

  • Mixing up which point is (x1,y1)(x_1, y_1) and which is (x2,y2)(x_2, y_2) in the section formula.
  • Plugging a guessed ratio into the x-formula without first solving for k from the y=0y = 0 condition.
  • Sign errors while handling the negative coordinates of both given points.

Real-world

  • This exact reasoning — finding where a segment crosses an axis — is used in computer graphics to detect where a line crosses a screen boundary.

Previously asked

2017Section C3 marksQ20

Find the ratio in which the x-axis divides the line segment joining the points (2, −3) and (5, 6). Also find the coordinates of the point of division.

Interesting facts

A positive value of kk (here k=5k = 5) always means internal division; a negative value would mean the axis is crossed only when the segment is extended (external division).

Since A(−6,5)A(-6, 5) lies above the x-axis and B(−4,−1)B(-4, -1) lies below it, the segment MUST cross the x-axis exactly once — the algebra simply pinpoints where.

Question details

Q29 of 53 questions

Board
CBSE
Class
Class 10
Subject
Mathematics (Basic)
Year
2026
Section
Section C
Marks
3
Type
Short Answer
Chapter
Coordinate Geometry
Topic
Section formula - ratio of division by x-axis
Bloom's level
apply
Syllabus topic
Unit 3: Coordinate Geometry — Section formula: ratio in which a point/axis divides a line segment and the point of division (CBSE Class 10 Mathematics)
Author
CBSE
Reviewed by
Padhantu Academic Team

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Frequently asked questions

Why do we set the y-coordinate equal to 0?

Every point lying on the x-axis has y=0y = 0. Since the point of division lies on the x-axis, its y-coordinate from the section formula must equal 0, and that single equation gives the ratio.

Why use k:1k : 1 instead of m:nm : n?

Using k:1k : 1 replaces two unknowns with one, simplifying the algebra. Once you find k, the ratio is simply k:1k : 1 (here 5:15 : 1).

How do we know the division is internal?

The value k=5k = 5 is positive, which indicates internal division. Geometrically it makes sense because one endpoint is above the x-axis and the other is below, so the axis cuts between them.