Question

A line making equal intercepts on coordinate axes and is tangent to the circle \(x^2+y^2 = 4\). The length of each intercept made by line on the coordinate axes is ...

• A. \(\sqrt{2}\)
• B. \(2\)
• C. \(2\sqrt{2}\)
• D. \(4\)

Hint:

A line with equal intercepts is x + y = c; its distance from the origin must equal the radius.

Answer 1

The Correct Option is C

Step 1: Understanding the Concept:
A line with equal intercepts \(c\) on both axes has the equation \(x+y=c\). It is tangent to a circle when its distance from the centre equals the radius.

Step 2: Apply the Condition:
The circle \(x^2+y^2=4\) has centre \((0,0)\) and radius 2. Distance from the origin to \(x+y-c=0\):
\[ \frac{|c|}{\sqrt{1^2+1^2}} = \frac{|c|}{\sqrt2} = 2 \Rightarrow |c| = 2\sqrt2 \]

Step 3: Read the Intercept:
The length of each intercept is \(|c|=2\sqrt2\).

Step 4: Check the Options:
\(\sqrt2\), 2 and 4 would give distances 1, \(\sqrt2\) and \(2\sqrt2\) from the origin, none equal to the radius 2. Only \(2\sqrt2\) works.

Final Answer:
Each intercept has length \(2\sqrt2\), option (C). \[ \boxed{\text{(C) } 2\sqrt{2}} \]