The locus of the mid-point of the chord of contact of tangents drawn from points lying on the straight line to the circle is
- A.
- B.
- C.
- D.
Show answer and solution
Correct option: A
Solution
Let a point on the line 4x - 5y = 20 be P(a, b), so 4a - 5b = 20.
The chord of contact of tangents from P(a,b) to x² + y² = 9 is: ax + by = 9.
Let the midpoint of this chord of contact be M(h, k). The chord with midpoint (h,k) on the circle x² + y² = 9 has equation: hx + ky = h² + k² (using T = S₁).
Since both equations represent the same chord:
a/h = b/k = 9/(h² + k²)
So a = 9h/(h² + k²) and b = 9k/(h² + k²).
Substituting into the constraint 4a - 5b = 20:
4·[9h/(h² + k²)] - 5·[9k/(h² + k²)] = 20
36h - 45k = 20(h² + k²)
Replacing (h,k) with (x,y): 20(x² + y²) - 36x + 45y = 0.
The chord of contact of tangents from P(a,b) to x² + y² = 9 is: ax + by = 9.
Let the midpoint of this chord of contact be M(h, k). The chord with midpoint (h,k) on the circle x² + y² = 9 has equation: hx + ky = h² + k² (using T = S₁).
Since both equations represent the same chord:
a/h = b/k = 9/(h² + k²)
So a = 9h/(h² + k²) and b = 9k/(h² + k²).
Substituting into the constraint 4a - 5b = 20:
4·[9h/(h² + k²)] - 5·[9k/(h² + k²)] = 20
36h - 45k = 20(h² + k²)
Replacing (h,k) with (x,y): 20(x² + y²) - 36x + 45y = 0.
[AIEEE 2012]