I am asked to find the center and intercepts of an ellipse and a hyperbola. also, to give the domain and range of each graph i am given.

## how do i find centers and intercepts of an ellipse and give the domain and range of a graph?

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# 1 Answer

Circle centered at origin: x^2 + y^2 = r^2

[Just use Pythagorean Theorem on any point (x,y) on the circle.]

Circle centered at (h,k): (x-h)^2 + (y-k)^2 = r^2

[Just translate circle centered at origin by <h,k>.]

Can also be: ((x-h)/r)^2 + ((y-k)/r)^2 = 1

Ellipse centered at (h,k): ((x-h)/a)^2 + ((y-k)/b)^2 = 1

[Dilated circle with different scale factors for x and y.]

Hyperbola centered at (h,k): ±((x-h)/a)^2 – ± ((y-k)/b)^2 = 1

Intercepts only make sense if they are either on the translated axes or on normal untranslated axes. We'll use the latter.

x-intercepts: set y = 0:

Circle: x^2 + 0^2 = r^2, x = ±r

Ellipse: (x/a)^2 + (0/b)^2 = 1, x = ±a

Hyperbola: ±(x/a)^2 – ± (0/b)^2 = 1

x/a = ±√(±1), x = ± a or ± ai

y-intercepts: set x = 0:

Circle: 0^2 + y^2 = r^2, y = ±r

Ellipse: (0/a)^2 + (y/b)^2 = 1, y = ±b

Hyperbola: ±(0/a)^2 – ± (y/b)^2 = 1

y/b = ±√(–±1), y = ± bi or ± b

Domain:

Circle: |x| <= r

Ellipse: |x| <= a

Hyperbola: |x| >= a or All real numbers.

Range:

Circle: |y| <= r

Ellipse: |y| <= b

Hyperbola: All real numbers or |y| >= b.

[Just use Pythagorean Theorem on any point (x,y) on the circle.]

Circle centered at (h,k): (x-h)^2 + (y-k)^2 = r^2

[Just translate circle centered at origin by <h,k>.]

Can also be: ((x-h)/r)^2 + ((y-k)/r)^2 = 1

Ellipse centered at (h,k): ((x-h)/a)^2 + ((y-k)/b)^2 = 1

[Dilated circle with different scale factors for x and y.]

Hyperbola centered at (h,k): ±((x-h)/a)^2 – ± ((y-k)/b)^2 = 1

Intercepts only make sense if they are either on the translated axes or on normal untranslated axes. We'll use the latter.

x-intercepts: set y = 0:

Circle: x^2 + 0^2 = r^2, x = ±r

Ellipse: (x/a)^2 + (0/b)^2 = 1, x = ±a

Hyperbola: ±(x/a)^2 – ± (0/b)^2 = 1

x/a = ±√(±1), x = ± a or ± ai

y-intercepts: set x = 0:

Circle: 0^2 + y^2 = r^2, y = ±r

Ellipse: (0/a)^2 + (y/b)^2 = 1, y = ±b

Hyperbola: ±(0/a)^2 – ± (y/b)^2 = 1

y/b = ±√(–±1), y = ± bi or ± b

Domain:

Circle: |x| <= r

Ellipse: |x| <= a

Hyperbola: |x| >= a or All real numbers.

Range:

Circle: |y| <= r

Ellipse: |y| <= b

Hyperbola: All real numbers or |y| >= b.