Length of the Chord – Circle
The length of the chord of the circle S = 0 having P (x₁, y₁) as its midpoint is 2 √|S₁₁|.
Proof: Let AB be the chord of the circle S = 0 having P as its midpoint.
Now PA = PB and PA. PB = |S₁₁|
⇒ PA² = |S₁₁|
⇒ PA = √|S₁₁|
Therefore, length of chord AB = 2 PA
= 2 √|S₁₁|
Example: Find the length of the chord of the circle x² + y² + 2x + 3y + 3 = 0 having (1, 2) as its mid-point.
Solution: Given that
x² + y² + 2x + 3y + 3 = 0
mid-point is P (1, 2)
we know that length of a chord is 2 √|S₁₁|
S₁₁ = x₁² + y₁² + 2x₁ + 3y₁ + 3 = 0
⇒ (1) + 4 + 2 + 6 + 3 = 0
⇒ S₁₁ = 16
Length of a chord is 2 √|S₁₁| = 2 √|16|
= 2 (4)
= 8.