# What curve does the equation (x-3)^2/4+(y-4)^2/9=1 represent and what are its points of intersection with the axes ?

Sep 7, 2017

This is an ellipse that does not intersect the axes...

#### Explanation:

Given:

${\left(x - 3\right)}^{2} / 4 + {\left(y - 4\right)}^{2} / 9 = 1$

Let's reduce the number of fractions we need to work with by multiplying both sides by $36$ first to get:

$9 {\left(x - 3\right)}^{2} + 4 {\left(y - 4\right)}^{2} = 36$

Subtracting $36$ from both sides and transposing, we get:

$0 = 9 {\left(x - 3\right)}^{2} + 4 {\left(y - 4\right)}^{2} - 36$

$\textcolor{w h i t e}{0} = 9 \left({x}^{2} - 6 x + 9\right) + 4 \left({y}^{2} - 8 y + 16\right) - 36$

$\textcolor{w h i t e}{0} = 9 {x}^{2} - 54 x + 81 + 4 {y}^{2} - 32 y + 64 - 36$

$\textcolor{w h i t e}{0} = 9 {x}^{2} + 4 {y}^{2} - 54 x - 32 y + 109$

We can find the intercepts with the $x$ axis by substituting $y = 0$, or equivalently covering up the terms involving $y$ to find:

$0 = 9 {x}^{2} - 54 x + 109$

$\textcolor{w h i t e}{0} = {\left(3 x\right)}^{2} - 2 \left(3 x\right) \left(9\right) + 81 + 28$

$\textcolor{w h i t e}{0} = {\left(3 x - 9\right)}^{2} + 28$

This has no real solutions, so there are no intercepts with the $x$ axis#.

We can find the intercepts with the $y$ axis by substituting $x = 0$, or equaivalently covering up the terms involving $x$ to find:

$0 = 4 {y}^{2} - 32 y + 109$

$\textcolor{w h i t e}{0} = {\left(2 y\right)}^{2} - 2 \left(2 y\right) \left(8\right) + 64 + 45$

$\textcolor{w h i t e}{0} = {\left(2 y - 8\right)}^{2} + 45$

This has no real solutions, so there are no intercepts with the $y$ axis.

Alternatively, we could have saved ourselves much of this algebra by noting that the equation:

${\left(x - 3\right)}^{2} / 4 + {\left(y - 4\right)}^{2} / 9 = 1$

is the standard form of the equation of an ellipse:

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

with centre $\left(h , k\right) = \left(3 , 4\right)$, semi minor axis of length $a = 2$ (in the $x$ direction) and semi major axis of length $b = 3$ (in the $y$ direction).

So the ellipse is $1$ unit from both axes...
graph{(x-3)^2/4+(y-4)^2/9=1 [-9, 11, -2.24, 7.76]}