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

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

Every point lying on the x-axis has $y = 0$. 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.

Coordinate Geometry

Section formula - ratio of division by x-axis

Official Answer

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

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

Marking Scheme

11 mark: assuming ratio $k : 1$ and correctly applying the section formula for the y-coordinate: $(-k + 5)/(k + 1) = 0$ .
21 mark: solving to get $k = 5$ , i.e. the ratio $5 : 1$ .
31 mark: substituting into the x-coordinate formula to obtain $x = -\frac{13}{3}$ and stating the point as $(-\frac{13}{3}, 0)$ .

Hint

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

Quick Oral Answer

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

Analysis & Explanation

Use the section formula with the x-axis condition $y = 0$ 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 $k : 1$ (a single unknown) and set the section-formula y-coordinate to 0.
Solving gives $k = 5$ , so the ratio is $5 : 1$ — since $k > 0$ , 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 $(-\frac{13}{3}, 0)$ .

Common mistakes

Mixing up which point is $(x_1, y_1)$ and which is $(x_2, y_2)$ in the section formula.
Plugging a guessed ratio into the x-formula without first solving for k from the $y = 0$ 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.

Common Mistakes

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

Previously Asked

2017Section CQ203 marks

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

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

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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 $y = 0$ . 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 $k : 1$ instead of $m : n$ ?

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

How do we know the division is internal?

The value $k = 5$ 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.

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