Find the ratio in which the line segment joining $A(1, -5)$ and $B(-4, 5)$ is divided by the $x$-axis. Also find the coordinates of the point of division.
Step-by-Step Solution
Key Concept: On $x$-axis, $y = 0 \Rightarrow \dfrac{k(5) + 1(-5)}{k + 1} = 0 \Rightarrow 5k = 5 \Rightarrow k = 1 \Rightarrow 1 : 1$. $x = \dfrac{1(-4) + 1(1)}{2} = -\dfrac{3}{2}$. Point is $\left(-\dfrac{3}{2}, 0\right)$.
Let ratio be $k : 1$. $y = \dfrac{5k - 5}{k + 1} = 0 \Rightarrow 5k = 5 \Rightarrow k = 1$. Ratio is $1 : 1$. [1.0 Mark]
$x = \dfrac{1(-4) + 1(1)}{2} = -\dfrac{3}{2}$. Point of division is $\left(-\dfrac{3}{2}, 0\right)$. [1.0 Mark]
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🎯 Official CBSE Marking Scheme:
Evaluating ratio $1 : 1$: 1.0 Mark
Finding coordinates $(-3/2, 0)$: 1.0 Mark
Correct Answer: