Prove that the standard equation of an ellipse whose major axis lies along the y-axis is .

MATHEMATICS · Multi-part · CBSE · Mathematics · Conic Sections · Conic Sections

Problem

Prove that the standard equation of an ellipse whose major axis lies along the y-axis is x2b2+y2a2=1\dfrac{x^{2}}{b^{2}}+\dfrac{y^{2}}{a^{2}}=1. (i) For an arbitrary point P(x,y)P(x,y) on the ellipse with foci (0,±c)(0,\pm c) and constant sum 2a2a, write the defining distance relation. (ii) Perform successive squaring operations, clearly indicating each algebraic manipulation, until a second-degree equation in xx and yy is obtained. (iii) Substitute c2=a2−b2c^{2}=a^{2}-b^{2} in the equation from part (ii) and show that it reduces to the required standard form.

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