An ellipse is tangent to both the $x$-axis and the $y$-axis, and its foci are at $(2, -3)$ and $(2, -8).$ Find the length of the major axis.
8 answers
Hey bot
Hello! How can I assist you today?
The major axis clearly lies on the line x = 2, with the center at (2,-11/2).
So a = 11/2 and the major axis has length 11.
So a = 11/2 and the major axis has length 11.
That's correct! Since the ellipse is tangent to both the $x$-axis and the $y$-axis, its center lies on the line $x=2$ and $y=-\frac{11}{2}.$ Therefore, the distance from the center to each focus is $c=\frac{5}{2}.$ The length of the major axis is $2a,$ where $a$ is the distance from the center to each vertex. Using the Pythagorean theorem, we have:
$$a^2+c^2= (\frac{11}{2})^2 + (\frac{5}{2})^2 = \frac{125}{4}$$
Since $a$ is positive, we take the positive square root to get $a=\frac{\sqrt{125}}{2}=\frac{5\sqrt{5}}{2}.$ Thus, the length of the major axis is $2a=2\cdot\frac{5\sqrt{5}}{2}=5\sqrt{5}.$
$$a^2+c^2= (\frac{11}{2})^2 + (\frac{5}{2})^2 = \frac{125}{4}$$
Since $a$ is positive, we take the positive square root to get $a=\frac{\sqrt{125}}{2}=\frac{5\sqrt{5}}{2}.$ Thus, the length of the major axis is $2a=2\cdot\frac{5\sqrt{5}}{2}=5\sqrt{5}.$
Find me a refrigerator that's 28 cubic feet, has a water dispenser and a French door, for under 2000$
One refrigerator that matches your criteria is the LG LFXS28968S. It has a capacity of 28 cubic feet, a water dispenser, and French doors. It is priced at around $1,900 on various online retailers such as Best Buy, Home Depot, and Amazon. Note that prices may vary based on location and availability.
Find me a leather, accent chair, for 200$
Certainly! Here is a leather accent chair that is priced around $200:
1. Rivet Andrews Contemporary Top-Grain Leather Accent Chair - This chair features a sleek and modern design with genuine top-grain leather upholstery. It has a comfortable cushioned seat and sturdy wooden legs. The price for this chair is approximately $200 and can be found on Amazon.
Please note that prices may vary depending on the retailer and any ongoing discounts or promotions. Additionally, availability may vary as well.
1. Rivet Andrews Contemporary Top-Grain Leather Accent Chair - This chair features a sleek and modern design with genuine top-grain leather upholstery. It has a comfortable cushioned seat and sturdy wooden legs. The price for this chair is approximately $200 and can be found on Amazon.
Please note that prices may vary depending on the retailer and any ongoing discounts or promotions. Additionally, availability may vary as well.