Find the ratio in which the line $2x + y - 4 = 0$ divides the line segment joining the points $A(2, -2)$ and $B(3, 7)$.
Step-by-Step Solution
Key Concept: Let ratio be $k : 1$. Use section formula to express coordinates of point $P$ in terms of $k$, then substitute $P$ into line equation $2x + y - 4 = 0$.
Stepwise Solution:
Let point $P = \left(\dfrac{3k + 2}{k + 1}, \dfrac{7k - 2}{k + 1}\right)$. [1.0 Mark]
Since $P$ lies on $2x + y - 4 = 0$:
$2\left(\dfrac{3k + 2}{k + 1}\right) + \dfrac{7k - 2}{k + 1} - 4 = 0$. [1.0 Mark]
$2(3k + 2) + 7k - 2 - 4(k + 1) = 0 \Rightarrow 6k + 4 + 7k - 2 - 4k - 4 = 0 \Rightarrow 9k - 2 = 0 \Rightarrow k = \dfrac{2}{9}$.
Required ratio is $2 : 9$. [1.0 Mark]
Marking Scheme:
• Expressing point $P(x,y)$ using section formula with ratio $k:1$: 1.0 Mark
• Substituting $P(x,y)$ into line equation $2x + y - 4 = 0$: 1.0 Mark
• Solving for $k = 2/9$ (ratio $2:9$): 1.0 Mark
Correct Answer: