Question

Find the ratio in which point P (k, 7) divides the segment joining A (8, 9) and B (1, 2). Also find k.

Answer 1

Hint: Use the internal section formula\[P(x,y) \equiv (\dfrac{{m{x_2} + n{x_1}}}{{m + n}},\dfrac{{m{y_2} + n{y_1}}}{{m + n}})\]and substitute the given values to obtain the equations for the coordinates of point P.
Convert the ratio $m:n$into $r:1$by dividing the numerator and denominator of the fractions in the RHS of the equations of the coordinates by n; and then replace \[\dfrac{m}{n}\]by r.
Find the value of r by using the equation of the y-coordinate of P to get the value of \[\dfrac{m}{n}\]. Substitute the values of m and n in the equation for k to find its value.

Complete step by step solution:
We are given three points P (k, 7), A (8, 9) and B (1, 2) such that the point P divides the segment joining the points A and B in some ratio.
We are asked to find this ratio.
Also, we can see that the x-coordinate of the point P is denoted by k.
This means that the value of k is unknown. So, we are also asked to find the value of k.
We will solve this question using the section formula.
Now, there are two types of section formula available to solve such questions based on whether the point divides the segment internally or externally.
But as we can see, in this case, it is not explicitly mentioned that the division is internal or external.
In such scenarios, we use the internal section formula which is as follows:
If a point$P \equiv (x,y)$divides the segment formed by joining the points$A \equiv ({x_1},{y_1})$and$B \equiv ({x_2},{y_2})$ in the ratio say$m:n$internally, then the coordinates of P are given by
\[P(x,y) \equiv (\dfrac{{m{x_2} + n{x_1}}}{{m + n}},\dfrac{{m{y_2} + n{y_1}}}{{m + n}})\]
The pictorial representation of the internal division is as follows:

Using this section formula and substituting for the given coordinates of the points P, A, and B, we have:
\[P(k,7) \equiv (\dfrac{{m + 8n}}{{m + n}},\dfrac{{2m + 9n}}{{m + n}})\]
On comparing the coordinates, we get
$k = \dfrac{{m + 8n}}{{m + n}}........(1)$
$7 = \dfrac{{2m + 9n}}{{m + n}}.......(2)$
But the ratio is not known, so what we can do is divide the numerator and denominator of the RHS of both the equations (1) and (2) by n
Thus, we get
$k = \dfrac{{m + 8n}}{{m + n}} = \dfrac{{\dfrac{m}{n} + 8}}{{\dfrac{m}{n} + 1}}$
\[7 = \dfrac{{2m + 9n}}{{m + n}} = \dfrac{{\dfrac{{2m}}{n} + 9}}{{\dfrac{m}{n} + 1}}\]
Now denote \[\dfrac{m}{n}\]by r.
Then the ratio becomes$r:1$
Then we get
$k = \dfrac{{m + 8n}}{{m + n}} = \dfrac{{r + 8}}{{r + 1}}$
\[7 = \dfrac{{2m + 9n}}{{m + n}} = \dfrac{{2r + 9}}{{r + 1}}\]
Now,
\[
  7 = \dfrac{{2r + 9}}{{r + 1}} \\
   \Rightarrow 7(r + 1) = 2r + 9 \\
   \Rightarrow 7r + 7 = 2r + 9 \\
   \Rightarrow 7r - 2r = 9 - 7 = 2 \\
   \Rightarrow 5r = 2 \\
   \Rightarrow r = \dfrac{2}{5} \\
 \]
Therefore, we get\[\dfrac{m}{n} = \dfrac{2}{5}\]
Hence the required ratio is$2:5$.
This implies that m = 2 and n = 5.
Using these values in equation (1), we get
$k = \dfrac{{2 + 8(5)}}{{2 + 5}} = \dfrac{{2 + 40}}{{2 + 5}} = \dfrac{{42}}{7} = 6$
Hence the value of k is 6.

Note: Dividing by \[\dfrac{m}{n}\] is necessary to reduce the number of variables for easier solving of equations.
Not converting the ratio of $m:n$ in the form of $r:1$ is the key to solving such problems where the ratio and the coordinates are not known to us.