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Question: A line passes through \((x_{1}, y_{1})\) and \((h, k)\). If the slope of the line is m, show that \(...

A line passes through (x1,y1)(x_{1}, y_{1}) and (h,k)(h, k). If the slope of the line is m, show that k−y1=m(h−x1)k - y_{1} = m(h - x_{1})

Explanation

Solution

Hint: The slope of a line is defined as the ratio change in y coordinates to the change in x coordinates. Mathematically, the slope between two points (x1,y1)(x_{1}, y_{1}) and (h,k)(h, k) can be represented as-
m=y2−y1x2−x1m = \dfrac{{{y_2} - {y_1}}}{{{x_2} - {x_1}}}
The coordinates of two points are given in the question, so we can prove the relation by first finding the slope.

Complete step-by-step answer:

We have been given two points (x1, y1) and (h, k). The slope of these lines is m. It can be easily calculated as-
m=k−y1h−x1m = \dfrac{{k - {y_1}}}{{h - {x_1}}}
Cross-multiplying the equation by (h−x1)(h - x_{1}), we get-
k−y1=m(h−x1)k - y_{1} = m(h - x_{1})
Hence, the given expression is verified.

Note: In questions similar to this, we just need to simply identify and apply the formula and simplify it to get the required expression. For example, in this question, we had to cross multiply the equation to get the required expression. We must make sure to use the right formula for finding the slope. If we interchange the x and y coordinates y mistake, then we will not be able to prove the right result.