Consider the ellipse ( x^ 2 a^ 2 + y^ 2 b^ 2 =1 )

MATHEMATICS · Subjective · CBSE · Mathematics · Conic Sections · Conic Sections

Problem

Consider the ellipse x2a2+y2b2=1\dfrac{x^{2}}{a^{2}}+\dfrac{y^{2}}{b^{2}}=1, where a>b>0a>b>0. (i) State the algebraic condition that a point P(x1,y1)P(x_{1},y_{1}) must satisfy in order to lie (A) on the ellipse, (B) inside the ellipse and (C) outside the ellipse. (ii) Explain, without performing any numerical calculation, how the sign of the expression x12a2+y12b2−1\dfrac{x_{1}^{2}}{a^{2}}+\dfrac{y_{1}^{2}}{b^{2}}-1 determines the exact position of the point with respect to the ellipse.

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