Q. 85.0( 3 Votes )

# If the minor axis of an ellipse subtends an equilateral triangle with vertex at one end of major axis, then write the eccentricity of the ellipse.

Answer :

Given that the minor axis of an ellipse subtends an equilateral triangle with vertex at one end of major axis. We need to find the eccentricity of the ellipse.

Let us assume that ends of minor axis be B = (0,b) and C(0, - b) and end of major axis be A(a,0)

We know that the distance between the ends of minor axis is 2b.

Let us find the distance AB

We know that the distance between the points (x_{1},y_{1}) and (x_{2},y_{2}) is .

⇒

⇒ .

We know that sides of an equilateral triangle are equal.

⇒ AB = 2b

⇒ AB^{2} = 4b^{2}

⇒ a^{2} + b^{2} = 4b^{2}

We know that b^{2} = a^{2}(1 - e^{2}),

⇒ a^{2} + a^{2} - a^{2}e^{2} = 4a^{2} - 4a^{2}e^{2}

⇒ 2a^{2} = 3a^{2}e^{2}

⇒

⇒ .

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