Question 5 of 70intermediate🔧 ApplyShort Answer3 marks

Find the slope of a line, which passes through the origin, and the mid-point of the line segment joining the points P(0,−4)P(0, -4) and B(8,0)B(8, 0).

Correct Answer

The slope of the line is −12-\frac{1}{2}.

To find the slope, we first determine the coordinates of the mid-point of the line segment joining P(0,−4)P(0, -4) and B(8,0)B(8, 0). Using the mid-point formula from the context, the coordinates (x,y)(x, y) are given by:

x=0+82=4x = \frac{0 + 8}{2} = 4 y=−4+02=−2y = \frac{-4 + 0}{2} = -2

So, the mid-point is (4,−2)(4, -2).

Now, we find the slope of the line passing through the origin (0,0)(0, 0) and the mid-point (4,−2)(4, -2). Using the slope formula m=y2−y1x2−x1m = \frac{y_2 - y_1}{x_2 - x_1}:

m=−2−04−0=−24=−12m = \frac{-2 - 0}{4 - 0} = \frac{-2}{4} = -\frac{1}{2}

Thus, the required slope is −12-\frac{1}{2}.

Exercise: EXERCISE 9.1 | Q: 5 | (Page 8)
For More Understanding

Explanation

The solution follows the method outlined in the provided text. First, Section III (Mid-point formula) is used to find the coordinate connecting the two given points. Then, the definition of slope provided in Example 1 is used to find the gradient between the origin and the calculated mid-point.

Solution Steps

  1. Step 1: Identify the given points as P(0,−4)P(0, -4) and B(8,0)B(8, 0). The line passes through the origin (0,0)(0, 0).

  2. Step 2: Calculate the mid-point of the segment joining PP and BB using the formula (x1+x22,y1+y22)\left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right). This yields (0+82,−4+02)=(4,−2)\left( \frac{0+8}{2}, \frac{-4+0}{2} \right) = (4, -2).

  3. Step 3: Calculate the slope of the line joining the origin (0,0)(0, 0) and the mid-point (4,−2)(4, -2) using the formula m=y2−y1x2−x1m = \frac{y_2 - y_1}{x_2 - x_1}.

  4. Step 4: Substitute the values: m=−2−04−0=−24=−12m = \frac{-2 - 0}{4 - 0} = \frac{-2}{4} = -\frac{1}{2}.

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