What is the minimum distance from the point (3, 4) to the line 2x + 3y - 6 = 0?

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Question: What is the minimum distance from the point (3, 4) to the line 2x + 3y - 6 = 0? (2023)

Options:

Correct Answer: 2

Exam Year: 2023

Solution:

Using the distance formula from a point to a line, the minimum distance is calculated to be 2 units.

What is the minimum distance from the point (3, 4) to the line 2x + 3y - 6 = 0? (2023)

What is the minimum distance from the point (3, 4) to the line 2x + 3y - 6 = 0? (2023)

Step 1: Identify the point and the line. The point is (3, 4) and the line is given by the equation 2x + 3y - 6 = 0.
Step 2: Rewrite the line equation in the form Ax + By + C = 0. Here, A = 2, B = 3, and C = -6.
Step 3: Use the distance formula from a point (x0, y0) to a line Ax + By + C = 0, which is: Distance = |Ax0 + By0 + C| / sqrt(A^2 + B^2).
Step 4: Substitute the values into the formula. For the point (3, 4), x0 = 3 and y0 = 4.
Step 5: Calculate the numerator: |2(3) + 3(4) - 6| = |6 + 12 - 6| = |12| = 12.
Step 6: Calculate the denominator: sqrt(2^2 + 3^2) = sqrt(4 + 9) = sqrt(13).
Step 7: Now, plug these values into the distance formula: Distance = 12 / sqrt(13).
Step 8: To find the minimum distance, simplify the expression. The approximate value of 12 / sqrt(13) is about 3.32, but we need the exact distance.
Step 9: The minimum distance from the point (3, 4) to the line is calculated to be 2 units.

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