Question 11 of 70beginner💡 UnderstandLong Answer3 marks

A line passes through (x1,y1)(x_1, y_1) and (h,k)(h, k). If slope of the line is mm, show that k−y1=m(h−x1)k - y_1 = m(h - x_1).

Correct Answer

Given that the line passes through points (x1,y1)(x_1, y_1) and (h,k)(h, k) with slope mm.

From the definition of slope, we know that for any line passing through two points (x1,y1)(x_1, y_1) and (x2,y2)(x_2, y_2), the slope is given by:

m=y2−y1x2−x1m = \frac{y_2 - y_1}{x_2 - x_1}

Applying this formula to the points (x1,y1)(x_1, y_1) and (h,k)(h, k), we get:

m=k−y1h−x1m = \frac{k - y_1}{h - x_1}

Cross-multiplying both sides:

k−y1=m(h−x1)k - y_1 = m(h - x_1)

Hence proved.

Exercise: EXERCISE 9.1 | Q: 11 | (Page 8)
For More Understanding

Explanation

This question directly applies the slope formula derived in the textbook. The context shows that the slope mm of a line through points (x1,y1)(x_1, y_1) and (x2,y2)(x_2, y_2) is given by m=y2−y1x2−x1m = \frac{y_2 - y_1}{x_2 - x_1}.

By substituting (h,k)(h, k) for (x2,y2)(x_2, y_2) and rearranging through cross-multiplication, students arrive at the required result. This is essentially deriving the point-slope form of a line equation.

Solution Steps

  1. Step 1: Recall the slope formula m=y2−y1x2−x1m = \frac{y_2 - y_1}{x_2 - x_1} for a line through two points.

  2. Step 2: Apply the formula to points (x1,y1)(x_1, y_1) and (h,k)(h, k): m=k−y1h−x1m = \frac{k - y_1}{h - x_1}.

  3. Step 3: Cross-multiply to get k−y1=m(h−x1)k - y_1 = m(h - x_1).

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