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If the locous of the point of intersection of perpendicular tangent to the ellipse x 2 /a 2 +y 2 /b^2=1 is a circle with centre at(0,0) then the radius of circle is

If the locous of the point of intersection of perpendicular tangent to the ellipse x2/a2+y2/b^2=1 is a circle with centre at(0,0) then the radius of circle is
 

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1 Answers

Arun
25763 Points
2 years ago
 

The line y = mx ±√(a2 m2+b2) is a tangent to the given ellipse for all m. Suppose it passes through (h, k).

⇒ k - mh = √(a2m+ b2 ) ⇒ k2 + m2h2 - 2hkm = a2m2 + b2

⇒ m2 (h2 - a2) - 2hkm + k2 - b2 = 0.

If the tangents are at right angles, then m1m2 = -1.

⇒ (k2-b2)/(h2-a2 ) = - 1 ⇒ h2 + k2 = a2 + b2.

Hence the locus of the point (h, k) is x2 + y2 = a2 + b2 which is a circle. This circle is called the Director Circle of the ellipse.

 

hence radius = sqrt(a^2 + b^2)

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