The equation of the straight line whose slope is \( -\frac{2}{3} \) and which divides the line segment joining \( (1,2) \) and \( (-3,5) \) in the ratio 4:3 externally is:
Show Hint
- \( 2x + 3y - 12 = 0 \)
- \( 3x + 2y + 27 = 0 \)
- \( 2x + 3y - 9 = 0 \)
- \( 2x + 3y + 12 = 0 \)
\
The Correct Option is A
Solution and Explanation
Step 1: Find the externally dividing point
Using the section formula for external division, the coordinates of the point \( P(x,y) \) dividing the segment joining \( A(1,2) \) and \( B(-3,5) \) in the ratio \( m:n = 4:3 \) externally is given by: \[ x = \frac{m x_2 - n x_1}{m - n}, \quad y = \frac{m y_2 - n y_1}{m - n} \] Substituting values: \[ x = \frac{4(-3) - 3(1)}{4 - 3} = \frac{-12 - 3}{1} = -15. \] \[ y = \frac{4(5) - 3(2)}{4 - 3} = \frac{20 - 6}{1} = 14. \] So, the required point is \( (-15,14) \). Step 2: Find the equation of the line with given slope
The equation of a straight line with slope \( m = -\frac{2}{3} \) passing through \( (-15,14) \) is given by: \[ y - y_1 = m (x - x_1) \] Substituting: \[ y - 14 = -\frac{2}{3} (x + 15) \] Multiplying throughout by 3 to eliminate fraction: \[ 3(y - 14) = -2(x + 15) \] \[ 3y - 42 = -2x - 30 \] \[ 2x + 3y - 12 = 0 \] Step 3: Conclusion
Thus, the correct answer is: \[ \mathbf{2x + 3y - 12 = 0} \] \bigskip
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