Find the angle between the lines $2 x + 3 y - 5 = 0$ and $4 x - y + 7 = 0$ .

hard 3 marks Class 9 · Mathematics · Lines and Angles

Answer

≈70°

The correct answer is ≈70° — Step 8: Thus the angle between the two lines is approximately $7 0^{c i r c}$ .

Step-by-step solution

Step 1: Find the slope of the first line: $2 x + 3 y - 5 = 0; \Rightarrow; y = - \frac{2}{3} x + \frac{5}{3}$ , so $m_{1} = - \frac{2}{3}$ .
Step 2: Find the slope of the second line: $4 x - y + 7 = 0; R i g h t a r r o w; y = 4 x + 7$ , so $m_{2} = 4$ .
Step 3: Use the formula for the tangent of the angle between two lines: $tan θ = \frac{m _{2} - m _{1}}{1 + m _{1} m _{2}}$ .
Step 4: Compute the numerator: $m_{2} - m_{1} = 4 - (- \frac{2}{3}) = \frac{14}{3}$ .
Step 5: Compute the denominator: $1 + m_{1} m_{2} = 1 + (- \frac{2}{3}) \cdot 4 = 1 - \frac{8}{3} = - \frac{5}{3}$ .
Step 6: $tan θ = \frac{\frac{14}{3}}{- \frac{5}{3}} = \frac{14}{5}$ .
Step 7: Find $θ$ by taking the arctangent: $θ = arctan (\frac{14}{5}) \approx 7 0^{\circ}$ .
Step 8: Thus the angle between the two lines is approximately $7 0^{c i r c}$ .

Common mistake: Using a negative sign in the denominator incorrectly

Concept tested: Angle between two lines