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Question: Minimum area of the triangle by any tangent to the ellipse \(\frac{x^{2}}{a^{2}}\)+ \(\frac{y^{2}}{b...

Minimum area of the triangle by any tangent to the ellipse x2a2\frac{x^{2}}{a^{2}}+ y2b2\frac{y^{2}}{b^{2}}= 1 with the coordinate axes is-

A

a2+b22\frac{a^{2} + b^{2}}{2}

B

(a+b)22\frac{(a + b)^{2}}{2}

C

ab

D

(a−b)22\frac{(a - b)^{2}}{2}

Answer

ab

Explanation

Solution

Equation of tangent at point P Ž T = 0

Ž xa\frac{x}{a}cos q +yb\frac{y}{b}sin q = 1; R{acos⁡θ,0}\left\{ \frac{a}{\cos\theta},0 \right\}

& Q {0,bsin⁡θ}\left\{ 0,\frac{b}{\sin\theta} \right\}

Area of DCQR =12\frac{1}{2}(bsin⁡θ)\left( \frac{b}{\sin\theta} \right)=ab∣sin⁡2θ∣\frac{ab}{|\sin 2\theta|}

Ž Area (minimum) = ab