4.3 Q-1

Question Statement

Find the slope and angle of inclination of the line joining the given points and sketch the line in each case.

i. Points: $(-2, 4)$ and $(5, 11)$

ii. Points: $(-2, 3)$ and $(2, 7)$

iii. Points: $(4, 6)$ and $(4, 8)$

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Background and Explanation

In this problem, we are required to find the slope and the angle of inclination of the line joining two points.

• Slope is a measure of the steepness of a line and can be calculated using the formula:

$$m = \frac{y_2 - y_1}{x_2 - x_1}$$

• The angle of inclination $\theta$ of the line is the angle it makes with the positive x-axis. The relationship between the slope and the angle of inclination is given by:

$$m = \tan(\theta)$$

To find the angle, we take the inverse tangent (or arctangent) of the slope:

$$\theta = \tan^{-1}(m)$$

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Solution

i. Points: $(-2, 4)$ and $(5, 11)$

1. Find the slope of the line:

Using the slope formula:

$$m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{11 - 4}{5 - (-2)} = \frac{7}{7} = 1$$

2. Find the angle of inclination:

Since $m = 1$, we have:

$$\tan(\theta) = 1 \Rightarrow \theta = \tan^{-1}(1) = 45^\circ$$

Thus, the slope of the line is 1, and the angle of inclination is $45^\circ$.

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ii. Points: $(-2, 3)$ and $(2, 7)$

1. Find the slope of the line:

Using the slope formula:

$$m = \frac{7 - 3}{2 - (-2)} = \frac{4}{4} = 1$$

2. Find the angle of inclination:

Since $m = -9$, we have:

$$\tan(\theta) = -9 \Rightarrow \theta = \tan^{-1}(-9) = -83.66^\circ$$

Since the angle of inclination is negative, we adjust the result:

$$\theta = 180^\circ - 83.66^\circ = 96.34^\circ$$

Thus, the slope of the line is -9, and the angle of inclination is $96.34^\circ$.

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iii. Points: $(4, 6)$ and $(4, 8)$

1. Find the slope of the line:

Using the slope formula:

$$m = \frac{8 - 6}{4 - 4} = \frac{2}{0}$$

Since dividing by zero gives an undefined result, we conclude that the slope is undefined.

2. Find the angle of inclination:

For an undefined slope, the line is vertical, and the angle of inclination is always:

$$\theta = 90^\circ$$

Thus, the slope is undefined, and the angle of inclination is $90^\circ$.

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Key Formulas or Methods Used

• Slope formula:

$$m = \frac{y_2 - y_1}{x_2 - x_1}$$

• Angle of inclination:

$$m = \tan(\theta) \Rightarrow \theta = \tan^{-1}(m)$$

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Summary of Steps

1. Find the slope using the formula:

$$m = \frac{y_2 - y_1}{x_2 - x_1}$$

2. Find the angle of inclination by applying:

$$\theta = \tan^{-1}(m)$$

3. For vertical lines, the slope is undefined and the angle of inclination is $90^\circ$.

4. Sketch the lines for each case.

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