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Ramsey

Answered question

2020-11-12

Find and calculate the center, foci, vertices, asymptotes, and radius, as appropriate, of the conic sections

x^{2} + 2 y^{2} - 2 x - 4 y = - 1

Answer & Explanation

berggansS

Skilled2020-11-13Added 91 answers

Given

x^{2} + 2 y^{2} - 2 x - 4 y = - 1

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

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

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

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

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

It is in the form of

\frac{(x - h)^{2}}{a^{2}} + \frac{(y - k)^{2}}{b^{2}} = 1

a^{2} = 2, b^{2} = 1

a = \sqrt{2}, b = 1

Distance between center and focus

c = \sqrt{2 - 1}

= 1

Hence Center

(h, k) = (1, 1)

Foci

= (h \pm c, k)

= (1 \pm 1, 1)

Vertices

= (h \pm a, k)

= (1 \pm \sqrt{2}, 1)

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