Find the distance from the point (3, 4) to the line 2x + 3y - 6 = 0.

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Question: Find the distance from the point (3, 4) to the line 2x + 3y - 6 = 0.

Options:

Correct Answer: 3

Solution:

Distance = |Ax1 + By1 + C| / sqrt(A^2 + B^2) = |2*3 + 3*4 - 6| / sqrt(2^2 + 3^2) = |6 + 12 - 6| / sqrt(13) = 12 / sqrt(13).

Find the distance from the point (3, 4) to the line 2x + 3y - 6 = 0.

Find the distance from the point (3, 4) to the line 2x + 3y - 6 = 0.

Step 1: Identify the point (x1, y1) which is (3, 4).
Step 2: Write down the equation of the line in the form Ax + By + C = 0. Here, A = 2, B = 3, and C = -6.
Step 3: Use the distance formula: Distance = |Ax1 + By1 + C| / sqrt(A^2 + B^2).
Step 4: Substitute the values into the formula: Distance = |2*3 + 3*4 - 6| / sqrt(2^2 + 3^2).
Step 5: Calculate the numerator: 2*3 = 6, 3*4 = 12, so 6 + 12 - 6 = 12.
Step 6: Calculate the denominator: sqrt(2^2 + 3^2) = sqrt(4 + 9) = sqrt(13).
Step 7: Put the values together: Distance = 12 / sqrt(13).

Distance from a Point to a Line – This concept involves using the formula for calculating the perpendicular distance from a point to a line given in standard form.
Line Equation in Standard Form – Understanding how to identify coefficients A, B, and C from the line equation Ax + By + C = 0.
Absolute Value and Square Root – Applying absolute value and square root operations correctly in the context of distance calculation.