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Mathematics · Ch 7 — Conic Sections

Director Circle: Locus of Perpendicular Tangents (Hyperbola)

7.3.7

Director Circle: Locus of Perpendicular Tangents (Hyperbola)

Setting the two tangent slopes' product m1m2=−1m_1m_2=-1 (perpendicular tangents), using m1m2=y12+b2x12−a2m_1m_2=\dfrac{y_1^2+b^2}{x_1^2-a^2} from section 7.3.6:

y12+b2x12−a2=−1  ⟹  y12+b2=−(x12−a2)  ⟹  x12+y12=a2−b2.\dfrac{y_1^2+b^2}{x_1^2-a^2}=-1 \;\Longrightarrow\; y_1^2+b^2=-(x_1^2-a^2) \;\Longrightarrow\; x_1^2+y_1^2=a^2-b^2.

So the locus is

x2+y2=a2−b2(a>b),\boxed{x^2+y^2=a^2-b^2} \qquad (a>b), …

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