Question

M is a point on y-axis at a distance of 4 units from x-axis and it lies below the x-axis. The distance of point M from point Q(3, 1) is:

• A. \( \sqrt{2} \, \text{units} \)
• B. \( \sqrt{24} \, \text{units} \)
• C. \( \sqrt{34} \, \text{units} \)
• D. \( \sqrt{60} \, \text{units} \)

Hint:

Always interpret vertical distance from x-axis as y-coordinate. Use Pythagoras theorem for distance formula.

Answer 1

The Correct Option is C

Step 1: Identify the coordinates of the points.
The point M lies on the y-axis and 4 units below the x-axis, so:
\[ M = (0, -4) \]
The point Q is given as:
\[ Q = (3, 1) \]
Step 2: Use the distance formula to calculate the distance between M and Q:
\[ \text{Distance} = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \]
Substitute the values \( (x_1, y_1) = (0, -4) \) and \( (x_2, y_2) = (3, 1) \):
\[ \text{Distance} = \sqrt{(3 - 0)^2 + (1 - (-4))^2} = \sqrt{(3)^2 + (5)^2} = \sqrt{9 + 25} = \sqrt{34} \]
Step 3: Double-check the interpretation.
M is on the y-axis and 4 units below the x-axis, which means its y-coordinate is negative (−4). So M = (0, −4) is correct.
The calculation is also correct: \[ \sqrt{(3)^2 + (5)^2} = \sqrt{9 + 25} = \sqrt{34} \]
Step 4: Compare with the given options.
None of the provided options match \( \sqrt{34} \).
Therefore, there might be a misprint in the options.
Final Answer:
\[ \boxed{\sqrt{34}} \]