CONCEPT:
Let A (x1, y1) and B (x2, y2) be the two given points and the point P (x, y) divide the line joining the points A and B in the ratio m : n, then
- Point of internal division is given as: \(\left( {x,\;y} \right) = \left( {\frac{{m{x_2} + n{x_1}}}{{m + n}},\frac{{m{y_2} + n{y_1}}}{{m + n}}} \right)\)
- Point of external division is given as:\(\left( {x,y} \right) = \left( {\frac{{m{x_2} - n{x_1}}}{{m - n}},\frac{{m{y_2} - n{y_1}}}{{m - n}}} \right)\)
CALCULATE:
Let the line y - x + 2 = 0 divides the line joining the points (3, -1) and (8, 9) in the ratio m : n internally.
As we know that, the point of internal division is given as: \(\left( {x,\;y} \right) = \left( {\frac{{m{x_2} + n{x_1}}}{{m + n}},\frac{{m{y_2} + n{y_1}}}{{m + n}}} \right)\)
Here, x1 = 3, y1 = - 1, x2 = 8 and y2 = 9.
So, the point of internal division is \(\left( {x,\;y} \right) = \left( {\frac{{8m + 3n}}{{m + n}},\frac{{9m - n}}{{m + n}}} \right)\)
∵ the point of internal division lies on the given line i.e (x, y) will satisfy the equation of line.
\(⇒ \left( {\frac{{9m - n}}{{m + n}}} \right) - \left( {\frac{{8m\; + 3n}}{{m + n}}} \right) + 2 = 0\)
⇒ (9m - n) - (8m + 3n) + 2 ⋅ (m + n) = 0
⇒ 3m = 2n
⇒ m : n = 2 : 3
Hence, option C is the correct answer.
Let y − x + 2 = 0 divide the line joining the points (3, −1) and (8, 9) at point P in the ratio k : 1
\[\therefore P \equiv \left( \frac{3 + 8k}{k + 1}, \frac{- 1 + 9k}{k + 1} \right)\]
P lies on the line y − x + 2 = 0
\[\therefore \frac{- 1 + 9k}{k + 1} - \frac{3 + 8k}{k + 1} + 2 = 0\]
\[ \Rightarrow - 1 + 9k - 3 - 8k + 2k + 2 = 0\]
\[ \Rightarrow 3k = 2\]
\[ \Rightarrow k = \frac{2}{3}\]
Hence, the line y − x + 2 = 0 divides the line joining the points (3, −1) and (8, 9) in the ratio 2 : 3