Question

Suppose the angle between the tangents drawn from \((0,0)\) to the circle \[ (x+\lambda)^2+(y+1)^2=\lambda^2 \] is \(\frac{\pi}{2}\). Then \(\lambda\) satisfies:

• A. \(\lambda^2=1\)
• B. \(\lambda=0\)
• C. \(\lambda^2=4\)
• D. \(\lambda^2=9\)

Hint:

For tangents drawn from an external point to a circle, if the angle between tangents is \(\theta\), then use \[ \sin\frac{\theta}{2}=\frac{r}{d}, \] where \(r\) is radius and \(d\) is the distance of the external point from the centre.

Answer 1

The Correct Option is A

Step 1: Identify the centre and radius of the given circle.
The circle is (x + λ)² + (y + 1)² = λ². Its centre is C(-λ, -1) and its radius is r = |λ|. The external point is the origin O(0, 0), so OC = √(λ² + 1).

Step 2: Relate the angle between tangents to the geometry.
If θ is the angle between the two tangents from O to the circle, then sin(θ/2) = r/OC. Here θ = π/2.

Step 3: Substitute and solve for λ.
sin(π/4) = |λ|/√(λ² + 1) → 1/√2 = |λ|/√(λ² + 1). Squaring: 1/2 = λ²/(λ² + 1) → λ² + 1 = 2λ² → λ² = 1.

Step 4: Final conclusion.
The condition reduces to λ² = 1.