Write equations of ellipses from properties - High school Geometry Exercises

Find the equation of the ellipse that has the following characteristics :

The center is \left(1{,}2\right)
A horizontal axis with length 6.
A vertical axis with length 4.

\dfrac{\left(x-1\right)^2}{36}+ \dfrac{\left(y-2\right)^2}{16}=1

\dfrac{\left(x-1\right)^2}{4}+ \dfrac{\left(y-2\right)^2}{9}=1

\dfrac{\left(x-1\right)^2}{16}+ \dfrac{\left(y-2\right)^2}{36}=1

\dfrac{\left(x-1\right)^2}{9}+ \dfrac{\left(y-2\right)^2}{4}=1

The standard formula of an ellipse is:

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

Where the length of the horizontal axis is 2a, the length of the vertical axis is 2b, and the center is \left(h,k\right).

Here:

The equation is:

\dfrac{\left(x-1\right)^2}{9}+ \dfrac{\left(y-2\right)^2}{4}=1

Find the equation of the ellipse that has the following characteristics :

The vertices are (5, 0) and (–1, 0).
The foci are (4, 0) and (0, 0).

\dfrac{x^2}{5}+ \dfrac{\left({y-2}\right)^2}{9}=1

\dfrac{\left(x-2\right)^2}{9}+ \dfrac{y^2}{5}=1

\dfrac{x^2}{9}+ \dfrac{\left({y-2}\right)^2}{5}=1

\dfrac{\left(x-2\right)^2}{5}+ \dfrac{y^2}{9}=1

The standard formula of an ellipse is:

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

Where the length of the horizontal axis is 2a, the length of the vertical axis is 2b, and the center is \left(h,k\right).

Here, the length of the horizontal axis equals:

5-\left(-1\right)=6

a=3

The center is midway between the two foci. Hence:

\left(h,k\right)=\left(2{,}0\right)

Each focus is 2 units from the center, thus c=2.

b^2=a^2-c^2

b^2=3^2-2^2=5

The equation is:

\dfrac{\left(x-2\right)^2}{9}+ \dfrac{y^2}{5}=1

Find the equation of the ellipse that has the following characteristics :

The center is \left(2,-3\right)
A horizontal axis with length 12
A vertical axis with length 6

\dfrac{\left(x-2\right)^2}{36}+ \dfrac{\left(y+3\right)^2}{9}=1

\dfrac{\left(x+1\right)^2}{25}+ \dfrac{\left(y-2\right)^2}{16}=1

\dfrac{\left(x-4\right)^2}{49}+ \dfrac{\left(y-2\right)^2}{9}=1

\dfrac{\left(x+2\right)^2}{36}+ \dfrac{\left(y-3\right)^2}{25}=1

The standard formula of an ellipse is:

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

Where the length of the horizontal axis is 2a, the length of the vertical axis is 2b, and the center is \left(h,k\right).

Here:

a=6
b=3
h=2
k=-3

The equation is:

\dfrac{\left(x-2\right)^2}{36}+ \dfrac{\left(y+3\right)^2}{9}=1

Find the equation of the ellipse that has the following characteristics :

The center is \left(-5,-3\right)
A horizontal axis with length 14
A vertical axis with length 8

\dfrac{\left(x-1\right)^2}{36}+ \dfrac{\left(y-2\right)^2}{4}=1

\dfrac{\left(x+5\right)^2}{49}+ \dfrac{\left(y+3\right)^2}{16}=1

\dfrac{\left(x-5\right)^2}{49}+ \dfrac{\left(y-3\right)^2}{16}=1

\dfrac{\left(x-1\right)^2}{36}+ \dfrac{\left(y-2\right)^2}{4}=1

The standard formula of an ellipse is:

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

Where the length of the horizontal axis is 2a, the length of the vertical axis is 2b, and the center is \left(h,k\right).

Here:

a=7
b=4
h=-5
k=-3

The equation is:

\dfrac{\left(x+5\right)^2}{49}+ \dfrac{\left(y+3\right)^2}{16}=1

Find the equation of the ellipse that has the following characteristics :

The center is \left(0{,}2\right)
A horizontal axis with length 4
A vertical axis with length 6

\dfrac{x^2}{16}+ \dfrac{\left(y-2\right)^2}{36}=1

\dfrac{\left(x-1\right)^2}{36}+ \dfrac{y^2}{16}=1

\dfrac{\left(x-4\right)^2}{9}+ \dfrac{\left(y-2\right)^2}{4}=1

\dfrac{\left(x-5\right)^2}{16}+ \dfrac{\left(y-2\right)^2}{9}=1

The standard formula of an ellipse is:

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

Where the length of the horizontal axis is 2a, the length of the vertical axis is 2b, and the center is \left(h,k\right).

Here:

The equation is:

\dfrac{x^2}{16}+ \dfrac{\left(y-2\right)^2}{36}=1

Find the equation of the ellipse that has the following characteristics :

The center is \left(-1{,}4\right).
The vertices are (-1,-3) and (-1,11).
A horizontal axis with length 8

\dfrac{\left(x+1\right)^2}{16}+ \dfrac{\left(y-4\right)^2}{49}=1

\dfrac{\left(x-1\right)^2}{25}+ \dfrac{\left(y+2\right)^2}{36}=1

\dfrac{\left(x-1\right)^2}{49}+ \dfrac{\left(y-2\right)^2}{16}=1

\dfrac{\left(x-1\right)^2}{4}+ \dfrac{\left(y+4\right)^2}{9}=1

The standard formula of an ellipse is:

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

Where the length of the horizontal axis is 2a, the length of the vertical axis is 2b, and the center is \left(h,k\right).

Here, we have:

a=4
h=-1
k=4

The length of the vertical axis is:

11-(-3)=14

So:

b=7

The equation is:

\dfrac{\left(x+1\right)^2}{16}+ \dfrac{\left(y-4\right)^2}{49}=1

Find the equation of the ellipse that has the following characteristics :

The center is \left(3,-5\right)
A horizontal axis with length 10
A vertical axis with length 18

\dfrac{\left(x-3\right)^2}{25}+ \dfrac{\left(y+5\right)^2}{81}=1

\dfrac{\left(x-7\right)^2}{16}+ \dfrac{\left(y+4\right)^2}{36}=1

\dfrac{\left(x+1\right)^2}{25}+ \dfrac{\left(y-2\right)^2}{81}=1

\dfrac{\left(x-8\right)^2}{36}+ \dfrac{\left(y+2\right)^2}{16}=1

The standard formula of an ellipse is:

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

Where the length of the horizontal axis is 2a, the length of the vertical axis is 2b, and the center is \left(h,k\right).

Here:

a=5
b=9
h=3
k=-5

The equation is:

\dfrac{\left(x-3\right)^2}{25}+ \dfrac{\left(y+5\right)^2}{81}=1