Question

The line \(2x+y = 4\) intersects X and Y axes at points A and B respectively. Let \(C(p,q)\) be the point such that the lines AC and BC are perpendicular. The distance of point C from the midpoint of segment AB is...

• A. \(\sqrt{2}\)
• B. \(\sqrt{5}\)
• C. \(2\sqrt{2}\)
• D. \(2\sqrt{5}\)

Hint:

AC and BC perpendicular means C lies on the circle with AB as diameter.

Answer 1

The Correct Option is B

Step 1: Find A and B:
Put \(y=0\) in \(2x+y=4\): \(x=2\), so \(A(2,0)\). Put \(x=0\): \(y=4\), so \(B(0,4)\).

Step 2: Use the angle in a semicircle:
If \(AC\perp BC\), then \(\angle ACB=90^{\circ}\), so \(C\) lies on the circle with \(AB\) as diameter.

Step 3: Find the distance:
The midpoint of \(AB\) is \(M(1,2)\), the centre of the circle. The distance \(CM\) equals the radius \(=\dfrac{AB}{2}\).
\[ AB=\sqrt{2^2+4^2}=\sqrt{20}=2\sqrt5,\quad CM=\sqrt5 \]

Step 4: Why the other options are wrong.
\(\sqrt2\) and \(2\sqrt2\) do not match half of AB. \(2\sqrt5\) is the full length AB, not the radius.

Final Answer:
The distance is square root of 5. \[ \boxed{\text{(B) }\sqrt5} \]