# Coordinate geometry: finding the ratio in which a line segment is divided by a line

The question is:

Determine the ratio in which the line $3x + 4y - 9 = 0$ divides the line segment joining the points $A(1,3)$ and $B(2,7)$.

When I tried solving the question using section formula, which is: If $P$ divides the line segment $A(x,y) B(p,q)$ in ratio $m:n$, then coordinates of $P$ are given by $$\left\lbrace \frac{mp + nx}{m+n}, \frac{mq + ny}{m+n} \right\rbrace.$$

I got the answer : $(-6) : 25$

which I think is wrong and I'm not able to confirm it. If someone could show me their solution, I'll be really grateful.

• That is the right answer.You can calculate the ratio by using the 'x' coordinate equation (or the 'y' coordinate equation) and then check for the correctness by substituting it in the 'y' coordinate equation (or the 'x' coordinate equation). – lsp Mar 13 '13 at 8:49
• The negative sign means the line does not pass between the points. – Macavity Mar 13 '13 at 10:06
• oh. I think then it must be dividing the line segment externally, now I get it. I was confused since this was the first time when I encountered a negative ratio. – Peeyush Kushwaha Mar 13 '13 at 11:48
• No Peeyush. Negative ratio implies the point is dividing the line segment externally. – lsp Mar 13 '13 at 11:59
• yeah, thats what I said – Peeyush Kushwaha Mar 25 '13 at 17:54

## 2 Answers

Hint-

Step 1 - Calculate the equation of line joining $(1,3)$ and $(2,7)$.

Step 2 - Find the point of intersection of the two lines.

Step 3 - Find the ratio using the above mentioned formula and you will get the ratio.

The answer that I got is $-6:25$.

Using your formula $x=1,y=3, p=2,q=9$

So, $P\left(\frac{2m+n}{m+n},\frac{9m+3n}{m+n}\right)$

Now, as the points $P,A,B$ are collinear, the are of the $\triangle PAB=0$

We can use this formula to calculate the area of of the triangle.