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Coordinate Geometry
NCERT Exemplar
CBSE
Grade 10

Question:

If $P(x, y)$ is any point on the line segment joining the points $A(a, 0)$ and $B(0, b)$, show that $\dfrac{x}{a} + \dfrac{y}{b} = 1$.

Step-by-Step Solution

Key Concept: Let $P(x,y)$ divide $AB$ in ratio $k : 1$. Use section formula to express $x$ and $y$ in terms of $a, b, k$, then compute $\dfrac{x}{a} + \dfrac{y}{b}$.
By section formula for ratio $k : 1$:
$x = \dfrac{k(0) + 1(a)}{k + 1} = \dfrac{a}{k + 1} \Rightarrow \dfrac{x}{a} = \dfrac{1}{k + 1}$. [1.0 Mark]
$y = \dfrac{k(b) + 1(0)}{k + 1} = \dfrac{kb}{k + 1} \Rightarrow \dfrac{y}{b} = \dfrac{k}{k + 1}$. [1.0 Mark]
Adding both equations: $\dfrac{x}{a} + \dfrac{y}{b} = \dfrac{1}{k + 1} + \dfrac{k}{k + 1} = \dfrac{1 + k}{k + 1} = 1$. Proved! [1.0 Mark]

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🎯 Official CBSE Marking Scheme:
Finding $x/a = 1/(k+1)$: 1.0 Mark
Finding $y/b = k/(k+1)$: 1.0 Mark
Adding to get $x/a + y/b = 1$: 1.0 Mark

Correct Answer:
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