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In what ratio does the line x-y-2=0 divide the line segment jo

Question

In what ratio does the line $x - y - 2 = 0$ divide the line segment joining the points $A (3, - 1)$ and $B (8, 9)$ ?

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Harper Discussed

In what ratio does the line

x - y - 2 = 0

divide the line segment joining the points

A (3, - 1)

and

B (8, 9)

13 mins ago

Discuss this question LIVE

13 mins ago

Text solutionVerified

Using the section formula, if a point

(x, y)

divides the line joining the points

(x_{1}, y_{1})

and

(x_{2}, y_{2})

in the ratio

m : n

, then

(x, y) = (\frac{m x _{2} + n x _{1}}{m + n}, \frac{m y _{2} + n y _{1}}{m + n})

Given: A line segment joining the points

A (3, - 1)

and

B (8, 9)

and another line

x - y - 2 = 0

Let a point

P (x, y)

on the given line

x - y - 2 = 0

divides the line segment AB in the ratio

m : n

Therefore,

x = \frac{m x _{2} + n x _{1}}{m + n} = \frac{m \times 8 + n \times 3}{m + n} = \frac{8 m + 3 n}{m + n}

and

y = \frac{m y _{2} + n y _{1}}{m + n} = \frac{m \times 9 + n \times ( - 1 )}{m + n}

= \frac{9 m - n}{m + n}

Since point P lies on

x - y - 2 = 0

, so

\frac{8 m + 3 n}{m + n} - \frac{9 m - n}{m + n} - 2 = 0

\frac{8 m + 3 n}{m + n} - \frac{9 m - n}{m + n} = 2

8 m + 3 n - 9 m + n = 2 m + 2 n

- m + 4 n = 2 m + 2 n

- m - 2 m = + 2 n - 4 n

- 3 m = - 2 n

\frac{m}{n} = \frac{- 2}{- 3} = \frac{2}{3}

The required ratio is

2 : 3

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Question Text	In what ratio does the line $x - y - 2 = 0$ divide the line segment joining the points $A (3, - 1)$ and $B (8, 9)$ ?
Answer Type	Text solution:1
Upvotes	150

In what ratio does the line x−y−2=0 divide the line segment joining the points A(3,−1) and B(8,9)?

Text solutionVerified

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In what ratio does the line $x - y - 2 = 0$ divide the line segment joining the points $A (3, - 1)$ and $B (8, 9)$ ?