I need help with the following questions please.
Graph the ellipse.
((x-1)^2)/(9)+ ((y-2)^2)/(4)=1
(4(x+1)^2)+9(y-2)^2=36
Start by converting your equations to standard form. The first one is already in standard form. For the second one, you need to divide all terms by 36 so that the left-hand side equals one.
Then, identify your center and foci. For more information on that, there is a very good website. (Unfortunately, I'm unable to post links.) It's called "Pauls Online Notes: Algebra - Ellipses." Typing that into Google should bring it right up.
If you have any questions, just ask.
You already have the equation in the form
(x-x')^2/a^2 + (y-y')^2/b^2 = 1
That form tells you all you need to know about what the ellipse looks like
The center of the ellipse is at x=x'= 2. The major axis is along the y=y' ine and has a half-length of a=3 since the denominator under (x-1)^2 is 3^2). The minor axis is along the x=1 line and has a half-length of b=2 (from the 2^2 in the denominator under (y-2)^2.
You should also try computing a few points yourself. Assume a value of x-1 between -3 and +3 (x between -2 and 4), and compute the corresponding value(s) of y to get points on the ellipse.
Correction: The center of the ellipse is at (x',y') = (1,2)
Sure, I can help you with that. To graph these ellipses, we can use the standard form of an ellipse equation, which is
((x-h)^2)/(a^2) + ((y-k)^2)/(b^2) = 1
Here, (h, k) represents the coordinates of the center of the ellipse, and 'a' and 'b' represent the lengths of the major and minor axes, respectively.
Let's start with the first equation:
((x-1)^2)/(9) + ((y-2)^2)/(4) = 1
From this equation, we can determine that the center of the ellipse is (1, 2) and the length of the major axis is 2a = 2 * 3 = 6 (so a = 3) and the length of the minor axis is 2b = 2 * 2 = 4 (so b = 2).
To graph the ellipse, we can start by plotting the center point at (1, 2). Then, we can draw a horizontal line segment of length 6 units on both sides of the center point (1 unit to the left and 5 units to the right) to represent the major axis. Next, we draw a vertical line segment of length 4 units above and below the center point (2 units above and 2 units below) to represent the minor axis. Finally, we can sketch the curve of the ellipse so that it intersects these horizontal and vertical line segments.
Moving on to the second equation:
4(x+1)^2 + 9(y-2)^2 = 36
From this equation, we can determine that the center of the ellipse is (-1, 2) and the length of the major axis is 2a = 6 (so a = 3) and the length of the minor axis is 2b = 4 (so b = 2).
To graph this ellipse, we can start by plotting the center point at (-1, 2). Then, we can draw a horizontal line segment of length 6 units on both sides of the center point (3 units to the left and 3 units to the right) to represent the major axis. Next, we draw a vertical line segment of length 4 units above and below the center point (2 units above and 2 units below) to represent the minor axis. Finally, we can sketch the curve of the ellipse so that it intersects these horizontal and vertical line segments.
Remember to label the center point and any other key points on the graph to help visualize the ellipse more accurately.