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Mathematics · Ch 7 — Conic Sections

Director Circle: Locus of Perpendicular Tangents (Ellipse)

7.2.7

Director Circle: Locus of Perpendicular Tangents (Ellipse)

Setting up the perpendicularity condition. From section 7.2.6, the two tangent slopes from P(x1,y1)P(x_1,y_1) satisfy m1m2=y12−b2x12−a2m_1m_2=\dfrac{y_1^2-b^2}{x_1^2-a^2}. If the two tangents are mutually perpendicular, then m1m2=−1m_1m_2=-1:

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 of PP (replacing (x1,y1)(x_1,y_1) by the general (x,y)(x,y)) is

x2+y2=a2+b2.\boxed{x^2+y^2=a^2+b^2}. …

Figure 7.22Perpendicular tangents meeting on the director circle

What this figure shows. Two mutually perpendicular tangent lines to the ellipse meeting at a point lying on the circle x2+y2=a2+b2x^2+y^2=a^2+b^2. This figure gives the reader a concrete visual reference for the geometric configuration described in the surrounding text, tying the abstract statement to a picture of the actual case. …

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