13. Given that \( (a+b, a),(a-b, 2 a) \) and \( (b, c) \) are collinear points, express \( c \) in terms of \( a \) and \( b \).

To express \( c \) in terms of \( a \) and \( b \) given that the points \( (a+b, a) \), \( (a-b, 2a) \), and \( (b, c) \) are collinear, we can use the concept of the slope of a line. The slope of a line passing through two points \( (x_1, y_1) \) and \( (x_2, y_2) \) is given by the formula: \[ \text{Slope} = \frac{y_2 - y_1}{x_2 - x_1} \] Since the points are collinear, the slope between any two points will be the same. Let's calculate the slope between the first two points \( (a+b, a) \) and \( (a-b, 2a) \): \[ \text{Slope} = \frac{2a - a}{a-b - (a+b)} = \frac{a}{-2b} = -\frac{a}{2b} \] Now, let's calculate the slope between the second point \( (a-b, 2a) \) and the third point \( (b, c) \): \[ \text{Slope} = \frac{c - 2a}{b - (a-b)} = \frac{c - 2a}{2b - a} \] Since the points are collinear, the slopes calculated above will be equal. Therefore, we can set up the equation: \[ -\frac{a}{2b} = \frac{c - 2a}{2b - a} \] Now, we can solve this equation to find the value of \( c \) in terms of \( a \) and \( b \). Solve the equation by following steps: - step0: Solve for \(c\): \(-\frac{a}{2b}=\frac{c-2a}{2b-a}\) - step1: Swap the sides: \(\frac{c-2a}{2b-a}=-\frac{a}{2b}\) - step2: Rewrite the expression: \(\frac{c-2a}{2b-a}=\frac{-a}{2b}\) - step3: Multiply both sides of the equation by \(2b-a:\) \(\frac{c-2a}{2b-a}\times \left(2b-a\right)=\frac{-a}{2b}\times \left(2b-a\right)\) - step4: Multiply the terms: \(c-2a=\frac{-a\left(2b-a\right)}{2b}\) - step5: Divide the terms: \(c-2a=\frac{-2ab+a^{2}}{2b}\) - step6: Move the constant to the right side: \(c=\frac{-2ab+a^{2}}{2b}+2a\) - step7: Add the terms: \(c=\frac{2ab+a^{2}}{2b}\) The value of \( c \) in terms of \( a \) and \( b \) is \( c = \frac{2ab+a^{2}}{2b} \).