Outside the segment

The idea

If P lies on the line AB but outside the segment AB, with PA : PB = m₁ : m₂, then P divides the segment AB externally in the ratio m₁ : m₂.

Three in a row

Three lamp posts A, B and P stand on one straight road. A and B are 2 blocks apart. P is 4 blocks from A and 2 blocks from B.

Where is P?

• Between A and B
• Beyond A, on the far side from B
• Beyond B, on the far side from A

Slide P along the line

The line through A and B goes on in both directions. Slide P along it, one step at a time. Count the steps from P to A and from P to B.

Dividing from outside

If P lies on the line AB but outside the segment AB, and PA : PB = m₁ : m₂, then P divides AB externally in the ratio m₁ : m₂.

When P is between A and B it divides AB internally, which is the section formula of this chapter. When P is outside AB, one distance is the other one plus AB: beyond B, PA = PB + AB, and beyond A, PB = PA + AB.

In the toy, P at 4 steps from A and 2 steps from B gives PA : PB = 4 : 2 = 2 : 1, with P beyond B. The ratio 2 : 1 also fits one point *inside* AB, so the words internally and externally tell you which point is meant.

The formula for the coordinates of an external point is studied in higher classes. In Class 10 you recognise such a point and use PA and PB.

Notes

A point P on the line AB, outside the segment AB, with PA : PB = m₁ : m₂, divides AB externally in the ratio m₁ : m₂. Inside AB it divides AB internally.

Check yourself

A(0, 0), B(2, 1) and P(6, 3) lie on one line, and P is outside AB. What is PA : PB?

A, B and P are on one line with AB = 4, PA = 7 and PB = 3. Where is P?

P is beyond B on the line AB. AB = 6 and PA : PB = 3 : 1. Find PB.

Answer: 3

PA = 3 × PB and PA = PB + AB = PB + 6. So 3 × PB = PB + 6, 2 × PB = 6, and PB = 3. (Then PA = 9.)

Is there a point P on the line AB, outside the segment AB, with PA : PB = 1 : 1?

• 2 : 3. That is PB : PA. P is farther from A than from B, so PA is the longer one.
• 3 : 1. PB is the distance from P to B, not the length of AB. A to B is 1 step and B to P is 2 steps, so PB = √20 = 2√5.
• 3 : 2 — correct. Yes! PA = √(6² + 3²) = √45 = 3√5 and PB = √(4² + 2²) = √20 = 2√5, so PA : PB = 3 : 2.
• Between A and B. If P were between A and B, PA + PB would be AB = 4. Here 7 + 3 = 10.
• Beyond B — correct. Yes! Beyond B, PA = PB + AB, and 7 = 3 + 4.
• Beyond A. Beyond A we would have PB = PA + AB. Here PB = 3 is shorter than PA = 7.
• Yes, just beyond B. Beyond B, PA = PB + AB, which is longer than PB. So PA : PB is never 1 : 1 there.
• Yes, the mid-point of AB. The mid-point is inside AB, not outside it.
• No: outside AB, PA and PB always differ by AB — correct. Yes! Beyond B, PA = PB + AB, and beyond A, PB = PA + AB. The two distances are never equal outside AB; 1 : 1 only fits the mid-point inside.