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Grade 12Analytical Geometry

what is the maximum number of concurrent normals to an ellipse?

Agniva Kolay

11 Years agoGrade 12

1 Answer

Jitender Singh

Approved Tutor Answer11 Years ago

Ans: 4
Hello student,
Please find answer to your question below
Equation of ellipse:
$\frac{x^{2}}{a^{2}} + \frac{y^{2}}{b^{2}} = 1$
Equation of normal:
$\frac{ax}{cos\theta } - \frac{by}{sin\theta } = a^{2}-b^{2}$
Let the normal passes through point (x₁, y₁)
$\frac{ax_{1}}{cos\theta } - \frac{by_{1}}{sin\theta } = a^{2}-b^{2}$ $\frac{ax_{1}(1+tan^{2}\frac{\theta }{2})}{1-tan^{2}\frac{\theta }{2} } - \frac{by_{1}(1+tan^{2}\frac{\theta }{2})}{2tan\frac{\theta }{2} } = a^{2}-b^{2}$
This polynomial equation in theta have four real & distinct roots.
So maximun four normal is possible.

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