How do I find the length of the major and minor axes of the ellipse $\frac{x^2}{49}+\frac{y^2}{25}=1$?

Question

How do I find the length of the major and minor axes of the equation $\frac{x^2}{49}+\frac{y^2}{25}$? For the vertices I got $(7,0)$ and $(-7,0)$ The foci: $(2\sqrt{6},0)$, $(-2\sqrt{6},0)$ Eccentricity: $\frac{2\sqrt{6}}{7}$ I'm not sure if those are correct, but I'm also confused on what the major and minor axes are. $a$ is $7$, and $b$ is $5$, but I don't know if those would be the right values.

Answer 1

There's a lot going on here. First, you're missing an $=1$ in your equations.

The major axis is the longest "diameter" you can draw in the ellipse. The minor axis is the shortest. $a=7$ is what people would call the length of the major SEMI-axis. So the length of the major axis is $14$.

It might help to plot the $x$ and $y$ intercepts (which you get by plugging in $y=0$ and $x=0$, respectively.) Those will be the endpoint of the axes you're looking for.

Answer 2

Based on the context that you provide, you're likely dealing with an ellipse whose equation is $${x^2\over49} + {y^2\over25} \color{red}{= 1}.$$ The standard-form of an ellipse whose center is at the origin is given by $${x^2\over a^2} + {y^2\over b^2} = 1,$$ where $a$ and $b$ are the length of the horizontal and vertical axes respectively. Depending on whether $a$ is bigger or $b$ is bigger determines which one is the major axis. In this case, $a = 7$ and $b = 5$ by simple square root. We clearly see that $a = 7$ is bigger; since that is associated with the $x$ variable, you know that you have horizontal major axis, and thus you have the following vertices of the ellipse: $$(\pm a, 0), (0,\pm b) \quad = \quad (\pm 7, 0), (\pm 5, 0).$$ Geometrically speaking, this just means we go out $7$ both left and right from the center, and $5$ up and down from the center. Being able to interpret the ellipse geometrically in effect makes these kinds of problems much easier, as you can help reduce the number of formulas that you need to memorize.

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This is especially true when you begin to shift the center to some general $(h,k)$; then the vertices become $(h \pm a, k), (h, k\pm b)$. This is a lot more "specific stuff" to remember, but if you interpret it geometrically, there is less math you have to do and less formulas required to memorize.