Question

the length of the sub tangent at any point theta on the ellipse x^2/a^2 +y^2/b^2=1 is

Answer 1

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Answer 2

The length of the sub tangent at any point θ on the ellipse is aSin²θ|Sec θ|

• we know that length of subtangent = |y1/m|

• from the diagram,From point(acos θ,bsin θ) x1 = acos θ and y1 = bsin θ

• we know the equation of ellipse,(x²/a)+(y²/b) = 1 and equation of tangent is (xx1/a²)+(y-y1/b²) = 1

•the point lies at x-axis, therefore,y=0.

• x = OB OB = a²/x1;OR = x1

• from the diagram,length of dub tangent is BR = OB-OR

=(a²/|x1|)-x1 where x1 = acos θ

= (a/|cos θ|)-|acosθ|

= a(1-Cos² θ)/|Cos θ|

= aSin² θ/Cos θ

= aSin² θ|Secθ|.