Question

If a circle passing through the point \((-1,0)\) touches \(y\)-axis at \((0,2)\), then the length of the chord of the circle along the \(x\)-axis is :

• A. \(\frac{3}{2}\)
• B. \(\frac{5}{2}\)
• C. 3
• D. 5

Hint:

Centre is (h, 2) with radius |h|; use the point (-1, 0) to find h, then intersect with y = 0.

Answer 1

The Correct Option is C

Step 1: Touching the y-axis at \((0,2)\) means centre \((h,2)\) and radius \(|h|\).
Step 2: Passing through \((-1,0)\): \((-1-h)^2+4=h^2\Rightarrow 1+2h+4=0\Rightarrow h=-\frac{5}{2}\). Radius \(=\frac{5}{2}\).
Step 3: On the x-axis (\(y=0\)): \(\left(x+\frac{5}{2}\right)^2=\frac{25}{4}-4=\frac{9}{4}\Rightarrow x=-1\) or \(-4\).
Step 4: Chord length \(=|-1-(-4)|=3\).
Answer: (3) 3