Find the distance between the point (3, 4) and the line 2x - 3y = 6 using the distance formula.
Answer
\[d = \frac{|2*3 - 3*4 + 6|}{\sqrt{2^2 + (-3)^2}} = \frac{|6 - 12 + 6|}{\sqrt{4 + 9}} = \frac{0}{\sqrt{13}} = 0\]
Step 1 ：\(Step 1: \) Write the equation of the line in slope-intercept form (y = mx + b).
Step 2 ：\[2x - 3y = 6 \rightarrow y = \frac{2}{3}x - 2\]
Step 3 ：\(Step 2: \) Use the distance formula to calculate the distance between the point and the line.
Step 4 ：\[d = \frac{|Ax_1 + By_1 + C|}{\sqrt{A^2 + B^2}}\]
Step 5 ：\(Where, \) (x_1, y_1) is the point (3, 4), and A, B, and C are the coefficients from the line equation.
Step 6 ：\[A = 2, B = -3, C = 6\]
Step 7 ：\(Substitute \) A, B, C, x_1, and y_1 into the distance formula.
Step 8 ：\[d = \frac{|2*3 - 3*4 + 6|}{\sqrt{2^2 + (-3)^2}} = \frac{|6 - 12 + 6|}{\sqrt{4 + 9}} = \frac{0}{\sqrt{13}} = 0\]
