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: 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.

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

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

Because the ratio k came out positive ($k=5$), 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.

Coordinate Geometry

Section Formula (Coordinate Geometry)

Official Answer

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

section formularatio of divisionx-axis intersectioncoordinate geometryinternal divisionpoint of intersection

Marking Scheme

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

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 $5:1$ , then substitute back to find the x-coordinate, giving the point $\left(-\frac{13}{3}, 0\right)$ .

Analysis & 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 $y = 0$ for k to get the ratio, then substitute k back to find x.
A above axis ( $y=5$ ) and B below ( $y=-1$ ) 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 $x = 0$ ) finds where the y-axis divides a segment, and underlies centroid/area-of-triangle problems.

Common Mistakes

1Setting up the section formula with the points in the wrong order (swapping A and B), which flips the sign of the resulting ratio.
2Forgetting 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.
3Arithmetic errors while simplifying $-\frac{26}{6}$ to $-\frac{13}{3}$ , or forgetting to state the final y-coordinate as 0 in the answer point.

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 $1:1$ .

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.

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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 ( $k=5$ ), 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.