A tangent to the ellipse x2-a2-y2-b2-1 cuts the axes in M and N- Then the least length of MN is

Equation of tangent at (θ) is x/a cos θ + y/b sin θ = 1 It meets axes at M=(a/cos θ,0) and N=(0,b/sin θ) ∴ MN = √(a^2 sec^2 θ + b^2 cosec^2 θ) = √(a^2 + b^2 + ...

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Standard XII

Mathematics

Parametric Form of Tangent: Ellipse

A tangent to ...

Question

A tangent to the ellipse $\frac{x^{2}}{a^{2}} + \frac{y^{2}}{b^{2}} = 1$ cuts the axes in M and N. Then the least length of MN is

a + b

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a - b

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a^{2} + b^{2}

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a^{2} - b^{2}

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Solution

The correct option is A a + b
Equation of tangent at $(θ)$ is
$\frac{x}{a} c o s θ + \frac{y}{b} s i n θ = 1 I t m e e t s a x e s a t M = (\frac{a}{c o s θ}, 0) a n d N = (0, \frac{b}{s i n θ}) ∴ M N = \sqrt{a^{2} s e c^{2} θ + b^{2} c o s e c^{2} θ} = \sqrt{a^{2} + b^{2} + a^{2} c o t^{2} θ + b^{2} t a n^{2} θ} ∴ M i n i m u m v a l u e o f a^{2} c o t^{2} θ + b^{2} t a n^{2} θ i s 2 a b (∵ A . M \geq G . M) ∴ M i n i m u m v a l u e o f M N = a + b$

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A tangent to the ellipse x2a2+y2b2=1 cuts the axes in M and N. Then the least length of MN is

The correct option is A a + bEquation of tangent at (θ) is xacosθ+ybsinθ=1It meets axes at M=(acosθ,0)and N=(0,bsin θ)∴MN=√a2sec2θ+b2cosec2θ=√a2+b2+a2cot2θ+b2tan2θ∴Minimum value of a2cot2θ+b2tan2θ is 2ab(∵A.M≥G.M)∴Minimum value of MN=a+b

A tangent to the ellipse $\frac{x^{2}}{a^{2}} + \frac{y^{2}}{b^{2}} = 1$ cuts the axes in M and N. Then the least length of MN is