# 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 ?

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

Given:

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

is the standard form of the equation of an ellipse:

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The equation represents an ellipse centered at (3, 4) with a horizontal axis length of 4 units and a vertical axis length of 6 units.

To find the points of intersection with the axes:

- For the x-axis: Set (y = 0) and solve for (x).
- For the y-axis: Set (x = 0) and solve for (y).

For the x-axis: [ \frac{{(x - 3)^2}}{4} + \frac{{(0 - 4)^2}}{9} = 1 ] [ \frac{{(x - 3)^2}}{4} + \frac{16}{9} = 1 ] [ \frac{{(x - 3)^2}}{4} = 1 - \frac{16}{9} ] [ \frac{{(x - 3)^2}}{4} = \frac{9}{9} - \frac{16}{9} ] [ \frac{{(x - 3)^2}}{4} = \frac{{-7}}{9} ] There are no real solutions since the right-hand side is negative.

For the y-axis: [ \frac{{(0 - 3)^2}}{4} + \frac{{(y - 4)^2}}{9} = 1 ] [ \frac{9}{4} + \frac{{(y - 4)^2}}{9} = 1 ] [ \frac{{(y - 4)^2}}{9} = 1 - \frac{9}{4} ] [ \frac{{(y - 4)^2}}{9} = \frac{4}{4} - \frac{9}{4} ] [ \frac{{(y - 4)^2}}{9} = \frac{{-5}}{4} ] There are no real solutions since the right-hand side is negative.

So, the ellipse does not intersect the x-axis or the y-axis in real numbers.

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When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

- How do you find the slope of the line through the points (1,-2), and (3,-8)?
- How do you tell whether the lines through the given points are parallel, perpendicular, or neither: (1,0), (7,4)?
- How do you graph # x = 4 #?
- What is the slope of the line passing through # (-6,12);(-11,7) #?
- How do you graph #y=2x-4#?

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