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If the points $A(a, 0), B(0, b)$ and $P(x, y)$ are collinear, using slopes, prove that



Given points are A(a,0),B(0,b) and P(x,y)

For three points to be collinear, the slope of all pairs must be equal, that is the slope of AB = slope of BP = slope of PA.

slope $=\left(\frac{\mathrm{y}_{2}-\mathrm{y}_{1}}{\mathrm{x}_{2}-\mathrm{x}_{1}}\right)$

Slope of $A B=\left(\frac{b-0}{0-a}\right)=\frac{b}{-a}$

Slope of BP $=\left(\frac{y-b}{x-0}\right)=\frac{y-b}{x}$

Slope of PA $=\left(\frac{y-0}{x-a}\right)=\frac{y}{x-a}$

Now Slope of AB = BP = PA


Using the first two equality

$\Rightarrow \frac{\mathrm{b}}{-\mathrm{a}}=\frac{\mathrm{y}-\mathrm{b}}{\mathrm{x}}$

$\Rightarrow \mathrm{bx}=-\mathrm{a}(\mathrm{y}-\mathrm{b})$

$\Rightarrow \mathrm{bx}=-\mathrm{ay}+\mathrm{ab}$

Dividing the equation by “ab”, We get


$\Rightarrow \frac{\mathrm{x}}{\mathrm{a}}+\frac{\mathrm{y}}{\mathrm{b}}=1$

Hence proved.


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