Circle – Director Circle and Its Equation
Director Circle: The locus of the point of intersection of two perpendicular tangents to given circle is known as Director Circle.
Equation of Director Circle:
Method (1): The equation of any tangent to the circle
x² + y² = a² is y = mx + a √ (1 + m²) … (1)
Let P (h, k) be the point of intersection of the tangents. Then P (h, k) lies on (1).
k = m h + a √ (1 + m²)
k – m h = a √ (1 + m²)
Squaring on both sides
(k – m h)² = a²(1 + m²)
k² + m²h² – 2mkh = a² + m²a²
Let two roots be m₁ and m₂.
The tangent perpendicular distance
m₁ m₂ = -1
(k² – a²)/(h² – a²) = -1
k² – a² = -h² + a²
h² + k² = 2a²
Hence, the locus of P (h, k) is x² + y² = 2 a².
Method (2): From figure, CRPQ is a square, therefore
CQ = CP cos 45⁰
2a² = h² + k²
x² + y² = 2a²
Which is required locus.
Note: The equation of director circle for the circle (x – h) ² + (y – k) ² = a² is given by (x – h) ² + (y – k) ² = 2a².
Example: Find the equation of director circle for the circle x ² + y ² = 16.
Solution: Given that
x ² + y ² = 16
x ² + y ² = 4²
a = 4
The equation of director circle is x ² + y ² = 2(4)
x ² + y ² = 8.