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Coordinate Geometry
RD Sharma
CBSE
Grade 10
Question:
If $P(x, y)$ is any point on the line joining $A(a, 0)$ and $B(0, b)$, prove that $\dfrac{x}{a} + \dfrac{y}{b} = 1$.
Step-by-Step Solution
Key Concept: Slope $AP =$ Slope $AB \Rightarrow \dfrac{y - 0}{x - a} = \dfrac{b - 0}{0 - a} = -\dfrac{b}{a} \Rightarrow a y = -b(x - a) \Rightarrow b x + a y = a b$. Divide by $a b \Rightarrow \dfrac{x}{a} + \dfrac{y}{b} = 1$.
Slope of $AP =$ Slope of $AB \Rightarrow \dfrac{y - 0}{x - a} = \dfrac{b}{-a}$. [1.0 Mark] $-a y = b(x - a) \Rightarrow b x + a y = a b$. [1.0 Mark] Dividing both sides by $a b$: $\dfrac{x}{a} + \dfrac{y}{b} = 1$. Proved! [1.0 Mark]
--- 🎯 Official CBSE Marking Scheme: Equating slopes: 1.0 Mark Forming equation $bx + ay = ab$: 1.0 Mark Dividing by $ab$ to conclude intercept form: 1.0 Mark
Correct Answer:
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