prove-that-the-curve-x-1-2-y-2-x-1-2-y-2-4-is-an-ellipse-and-find-its-semi-major-axis-and-semi-minor-axis-

Question Number 196723 by mr W last updated on 30/Aug/23

p r o v e t h a t t h e c u r v e

\sqrt{{(x - 1)}^{2} + y^{2}} + \sqrt{{(x + 1)}^{2} + y^{2}} = 4

i s a n e l l i p s e a n d f i n d i t s s e m i

m a j o r a x i s a n d s e m i m i n o r a x i s .

Answered by Frix last updated on 30/Aug/23

Squaring, transforming etc . leads to

\frac{x^{2}}{4} + \frac{y^{2}}{3} = 1

Testing with y^{2} = \frac{12 - 3 x^{2}}{4} in the given equation

gives

∣ x + 4 ∣ + ∣ x - 4 ∣= 8; true for - 4 ⩽ x ⩽ 4

Commented by mr W last updated on 30/Aug/23

��

Answered by mr W last updated on 30/Aug/23

v = \sqrt{{(x - 1)}^{2} + y^{2}}

u = \sqrt{{(x + 1)}^{2} + y^{2}}

u + v = 4

u^{2} + v^{2} = 2 (x^{2} + y^{2} + 1)

u^{2} - v^{2} = 4 x

\Rightarrow u - v = x

\Rightarrow u = \frac{4 + x}{2}

\Rightarrow v = \frac{4 - x}{2}

u^{2} + v^{2} = \frac{2 (16 + x^{2})}{4} = 2 (x^{2} + y^{2} + 1)

3 x^{2} + 4 y^{2} = 12

\Rightarrow \frac{x^{2}}{2^{2}} + \frac{y^{2}}{{(\sqrt{3})}^{2}} = 1

\Rightarrow e l l i p s e w i t h a = 2, b = \sqrt{3}

Facebook Tweet Pin