Question

Ends of the major axis of ellipse at (10, -1) and (-4, -1), and one of focus is (8, -1). What is the equation of the ellipse?

Answer 1

Task:

Ends of the major axis of ellipse at (10, -1) and (-4, -1), and one of focus is (8, -1). What is the equation of the ellipse?

Solution:

$$a ^ {2} = b ^ {2} + c ^ {2}$$ $$e = \frac {c}{a} = \sqrt {1 - \frac {b ^ {2}}{a ^ {2}}}$$ $$a = \frac {10 - (-4)}{2} = 7$$ $$c = a - (10 - 8) = 5$$ $$\frac {c}{a} = \frac {5}{7} = \sqrt {1 - \frac {b ^ {2}}{7 ^ {2}}}$$ $$b = 2 \sqrt {6}$$

Equation with the center at (0;0)

$$\frac {x ^ {2}}{a ^ {2}} + \frac {y ^ {2}}{b ^ {2}} = 1$$

Equation with the center at $\left(\frac{10 - 4}{2}; -1\right)$

$$\frac {(x - 3) ^ {2}}{a ^ {2}} + \frac {(y - 1) ^ {2}}{b ^ {2}} = 1$$ $$\frac {(x - 3) ^ {2}}{7 ^ {2}} + \frac {(y + 1) ^ {2}}{(2 \sqrt {6}) ^ {2}} = 1$$$$- \frac {1}{4 9} (x - 3) ^ {2} + \frac {1}{2 4} (y + 1) ^ {2} = 1$$

Answer:

$$\frac {(x - 3) ^ {2}}{7 ^ {2}} + \frac {(y + 1) ^ {2}}{(2 \sqrt {6}) ^ {2}} = 1$$