-2

And lies:

(i) On the same side of A as B does

(ii) On the opposite side of A as B does

Now I already know the section formula which is applied when the point is outside the line. I plug in the values and somehow I get different answers. The section formula is given as follows:

(A) When the point lies internally

$$x = \frac{m_1x_2 + m_2x_1}{m_1 + m_2} ,y= \frac{m_1y_2 + m_2y_1}{m_1 + m_2} $$ $Where\ m_1\ and\ m_2\ are \ the\ ratios\ which\ divide\ the\ line.$

(B) When the point lies externally

$$x = \frac{m_1x_2 - m_2x_1}{m_1 - m_2} ,y= \frac{m_1y_2 - m_2y_1}{m_1 - m_2} $$ $Where\ m_1\ and\ m_2\ are \ the\ ratios\ which\ divide\ the\ line.$

As far as I understand the ratio or m and n in the first case are going to 1:2 and in the second part they will be 2:1. But in the solution, the author divides it in 2:1 in the first case and 2:3 in the second case.

And I don't understand why and how? In the first case let's say B is 1 far from A, then P has to be 2 because it has to be twice as far from A as B is. Isn't it safe to say that the ratio then will be 1:2. And in the second case, P has the same condition is only changing sides which I think should simply flip the ratio. Hence being 2:1.

Am I wrong on the way I am assigning ratios or something?

Would appreciate any help on this.

Community
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Arkilo
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1 Answers1

1

When $P$ is on the same side of $A$ as is $B$,

same side

then $B$ is between $A$ and $P$, the total length of the segment is $AP = AB+BP = 2AB$, and we have $AB:AP = 1:2$. On the other hand, when $P$ is on the opposite side,

enter image description here

then it’s $A$ that’s between the other two points. The total length is $BP=AB+AP=3AB$ and $AP:BP = 2:3$.

amd
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  • So my ratio of the first one is correct right? Why did the author write it as 2:1 – Arkilo May 03 '19 at 21:35
  • @Arkilo I chose to write the ratio as $AB:AP$, i.e., the part to the whole. One could just as well write it in the opposite order. The correct order depends on what you’re planning to do with this value, i.e., on this “section formula” that you’ve not included explicitly in your question. – amd May 03 '19 at 22:05
  • My bad I'll add it – Arkilo May 04 '19 at 09:12