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Question
If three points A (h, 0), P (a, b) and B (0, k) lie on a line, show that: \[\frac{a}{h} + \frac{b}{k} = 1\].
Solution
The given points are A (h, 0), P (a, b) and B (0, k).
Thus, we have,
\[\text { Slope of AP } = \frac{b - 0}{a - h}\]
\[\text { Slope of BP } = \frac{b - k}{a - 0}\]
For the given points to be collinear, we must have,
Slope of AP = Slope of BP
\[\Rightarrow \frac{b - 0}{a - h} = \frac{b - k}{a - 0}\]
\[ \Rightarrow \frac{b}{a - h} = \frac{b - k}{a}\]
\[ \Rightarrow ab = ab - ak - bh + hk\]
\[ \Rightarrow ak + bh = hk\]
\[ \Rightarrow \frac{a}{h} + \frac{b}{k} = 1 \left[\text { On dividing both sides by hk } \right]\]
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