Definitions [1]
Definition: Director circle
The locus of the point of intersection of two perpendicular tangents to a given circle is known as the director circle.
For circle:
x² + y² = a²
Director circle is:
x² + y² = 2a²
Key Points
Key Points: Equation of a Circle in Different Forms
| Form | Condition / Description | Equation |
|---|---|---|
| Standard form | Centre (0, 0), radius r | x² + y² = r² |
| Centre-radius form | Centre (h, k), radius r | (x − h)² + (y − k)² = r² |
| General form | General equation of a circle | x² + y² + 2gx + 2fy + c = 0 Centre: (−g, −f) Radius: √(g² + f² − c) |
| Diameter form | Ends of diameter (x₁, y₁), (x₂, y₂) | (x − x₁)(x − x₂) + (y − y₁)(y − y₂) = 0 |
| Parametric (centre at origin) | — | x = r cosθ, y = r sinθ |
| Parametric (centre (h, k)) | — | x = h + r cosθ, y = k + r sinθ |
| Intercepts on Axes |
\[\mathrm{X-axis}=2\sqrt{g^2-c}\]\[\mathrm{Y-axis}=2\sqrt{f^2-c}\] |
Condition for Radius:
| Condition | Result |
|---|---|
| g² + f² − c > 0 | Radius is real → Circle is real |
| g² + f² − c = 0 | Radius = 0 → Point circle |
| g² + f² − c < 0 | Radius imaginary → No real circle |
Key Points: Equation of a Circle in some special cases
| Case | Condition | Equation |
|---|---|---|
| Touches X-axis | Radius = y-coordinate (k) | (x − h)² + (y − k)² = k² |
| Touches Y-axis | Radius = x-coordinate (h) | (x − h)² + (y − k)² = h² |
| Touches both axes | Centre (±a, ±a), radius = a | (x ± a)² + (y ± a)² = a² |
| Passes through the origin | — | (x − h)² + (y − k)² = h² + k² |
| General form (origin case) | — | x² + y² − 2hx − 2ky = 0 |
Key Points: Equation of Tangent and Condition of Tangency
For Standard Circle: x² + y² = a²
| Sr. No. | Description | Formula |
|---|---|---|
| i. | Tangent at a point (x₁, y₁) | xx₁ + yy₁ = a² |
| ii. | Parametric form of tangent at P(θ) | x cosθ + y sinθ = a |
| iii. | Condition of tangency for the line y = mx + c | \[\mathrm{c=\pm a~\sqrt{1+m^{2}}}\] |
| Point of contact | \[\left(\frac{-\mathrm{a}^{2}\mathrm{m}}{\mathrm{c}},\frac{\mathrm{a}^{2}}{\mathrm{c}}\right)\] | |
| iv. | Equation of tangent in terms of its slope m | \[y=\mathrm{m}x\pm\mathrm{a}\sqrt{1+\mathrm{m}^{2}}\] |
| v. | Length of tangent from the point (x₁, y₁) | \[\sqrt{S_{1}}=\sqrt{x_{1}^{2}+y_{1}^{2}-a^{2}}\] |
| vi. | Equation of the Director circle | x² + y² = 2a² |
For General Circle: x² + y² + 2gx + 2fy + c = 0
| Sr. No. | Description | Formula |
|---|---|---|
| i. | Tangent at a point (x₁, y₁) | xx₁ + yy₁ + g(x + x₁) + f(y + y₁) + c = 0 |
| ii. | Length of tangent from the point (x₁, y₁) | \[\sqrt{S_{1}}=\sqrt{x_{1}^{2}+y_{1}^{2}+2gx_{1}+2fy_{1}+c}\] |
Number of Common Tangents:
| Case | Diagram | No. of Tangents | Condition |
|---|---|---|---|
| Disjoint circles |  |
4 | d > r₁ + r₂ |
| Touch externally |  |
3 | d = r₁ + r₂ |
| Intersecting circles |  |
2 | d < r₁ + r₂ |
| Touch internally |  |
1 | d = \[\left|\mathbf{R}_{1}-\mathbf{R}_{2}\right|\] |
| Concentric circles |  |
0 | d = 0 |
Equation of a pair of tangents:
(x² + y² − a²)(x₁² + y₁² − a²) = (xx₁ + yy₁ − a²)²
