Subjects algebra

Distance Parallel 72C31D

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1. **Problem statement:** Find the distance between the two parallel lines given by the equations $$4x + 3y - 2 = 0$$ and $$4x + 3y + 8 = 0$$. 2. **Formula used:** The distance $d$ between two parallel lines $$Ax + By + C_1 = 0$$ and $$Ax + By + C_2 = 0$$ is given by: $$ d = \frac{|C_2 - C_1|}{\sqrt{A^2 + B^2}} $$ 3. **Identify coefficients:** For both lines, $A = 4$, $B = 3$, $C_1 = -2$, and $C_2 = 8$. 4. **Calculate numerator:** $$|C_2 - C_1| = |8 - (-2)| = |8 + 2| = 10$$ 5. **Calculate denominator:** $$\sqrt{A^2 + B^2} = \sqrt{4^2 + 3^2} = \sqrt{16 + 9} = \sqrt{25} = 5$$ 6. **Calculate distance:** $$d = \frac{10}{5} = 2$$ 7. **Answer:** The distance between the two lines is $2$. Therefore, the correct choice is (c) 2.