91Ó°ÊÓ

The x-intercepts of a graph are the points where the curve crosses the x-axis. This happens when the y-coordinate is zero. So, to find the x-intercepts, we need to set y to 0 in the given equation and solve for x. For the given ellipse equation \(\frac{(x-3)^{2}}{4}+\frac{(y-4)^{2}}{9}=1\), substitute y = 0 and solve for x. During the process of simplification, if we arrive at a negative value under the square root, it means there are no real x-intercepts. This is because in the real number system, the square root of a negative number does not exist.

Similarly, the y-intercepts of a graph are where the curve crosses the y-axis. This happens when the x-coordinate is zero. So, to find the y-intercepts, we set x to 0 in the given equation and solve for y. For the ellipse equation \(\frac{(x-3)^{2}}{4}+\frac{(y-4)^{2}}{9}=1\), substitute x = 0 and solve for y. As with x-intercepts, we will check for a negative value under the square root. If a negative value exists, then there are no real y-intercepts.

An ellipse is a set of points such that the sum of the distances from two fixed points (foci) is constant. The standard form of an ellipse equation centered at (h, k) is: \(\frac{(x-h)^{2}}{a^{2}} + \frac{(y-k)^{2}}{b^{2}} = 1\). This form helps in identifying the horizontal and vertical distances (semi-major and semi-minor axes). In the given problem, the equation \(\frac{(x-3)^{2}}{4}+\frac{(y-4)^{2}}{9}=1\) describes an ellipse centered at (3, 4) with semi-major axis length of 3 (since 9 = 3^2) and semi-minor axis length of 2 (since 4 = 2^2). Knowing this standard form helps in visualizing the graph and understanding why there are no real intercepts in this exercise due to the shifted and scaled nature of the ellipse.