Question

Let the tangents drawn from the origin to the circle,
x² + y² - 8x - 4y + 16 = 0 touch it at the points A and B. The (AB)² is equal to :

• A. \( {{56} \over 5} \)
• B. \( {{32} \over 5} \)
• C. \( {{52} \over 5} \)
• D. \( {{64} \over 5} \)

Hint:

Length of tangent from the origin is \(\sqrt{S_1}\); chord of contact length is \(2Lr/\sqrt{L^2+r^2}\) (or find the chord of contact line and use the perpendicular distance from the centre).

Answer 1

The Correct Option is D

Step 1: Circle \(x^2+y^2-8x-4y+16=0\): centre (4, 2), radius \(r=\sqrt{16+4-16}=2\).
Step 2: Square of tangent length from the origin: \(L^2=S_1=0+0-0-0+16=16\), so \(L=4\).
Step 3: Chord of contact length \(AB=\dfrac{2Lr}{\sqrt{L^2+r^2}}=\dfrac{2\cdot4\cdot2}{\sqrt{20}}=\dfrac{16}{\sqrt{20}}\).
Step 4: \((AB)^2=\dfrac{256}{20}=\dfrac{64}{5}\).
Answer: (D) \(\dfrac{64}{5}\)