Hi
I have this question regarding ellipses.
the equation is..
x^2/25 + y^2/169= 1
it asks for
major and minor axis (i got 10 and 26)
x-intercept y-intercept ((5,0)(0,13))
foci (-12,0)
points of intersection with line y= 1-2x
graph
graph the equation of the ellipses after it is translated according to [(x,y) --> (x-3, y+1)]
Hi there!
To find the major and minor axis of an ellipse, you need to know the lengths of the semi-major and semi-minor axes. The equation of the ellipse you provided is in standard form, which is in the form (x - h)^2 / a^2 + (y - k)^2 / b^2 = 1, where (h, k) is the center of the ellipse, and a and b are the lengths of the semi-major and semi-minor axes, respectively.
In your equation, x^2/25 + y^2/169 = 1, the center is at (0, 0). It implies that the semi-major axis length (a) is √25 = 5, and the semi-minor axis length (b) is √169 = 13. Hence, the major axis length is 2a = 2(5) = 10, and the minor axis length is 2b = 2(13) = 26. Thus, you are correct!
To find the x-intercepts, you can set y = 0 in the equation of the ellipse, resulting in x^2/25 = 1. By solving this equation, you can find the x-intercepts, which are (5, 0) and (-5, 0). Similarly, to find the y-intercepts, you can set x = 0, yielding y^2/169 = 1. Solving for y gives y = ±13, so the y-intercepts are (0, 13) and (0, -13). Consequently, you are also correct about the intercepts.
The foci can be found using the equation c^2 = a^2 - b^2, where c is the distance from the center to each focus. From the given equation, a^2 = 25 and b^2 = 169, so c^2 = 25 - 169 = -144. As c^2 is negative, it implies that the ellipse is imaginary and doesn't have real foci.
To find the points of intersection with the line y = 1 - 2x, you can substitute y with 1 - 2x in the equation of the ellipse. By solving this equation, you can find the x-values of the points of intersection. Once you have the x-values, substitute them back into the equation of the line to obtain the corresponding y-values.
To graph the given equation of the ellipse, you can plot the center (0, 0) and then use the lengths of the semi-major and semi-minor axes to draw the ellipse. Connect the two foci on the major axis with a line that goes through the center. You can also plot the intercepts and any additional points you found. Finally, sketch the ellipse based on the information you have.
To graph the translated equation of the ellipse, [(x, y) --> (x - 3, y + 1)], you need to shift the center of the original ellipse, (0, 0), three units to the left and one unit up, resulting in a new center at (-3, 1). The new equation becomes (x + 3)^2 / 25 + (y - 1)^2 / 169 = 1. Use the same process as before to graph the new ellipse.
I hope this explanation helps! Let me know if you have any more questions.
To find the major and minor axis of the ellipse, we can look at the coefficients of x^2 and y^2 in the equation. In this case, we have:
x^2/25 + y^2/169 = 1
The denominator of x^2 gives us the square root of 25, which is 5. So, the length of the major axis is 2 * 5 = 10.
Similarly, the denominator of y^2 gives us the square root of 169, which is 13. So, the length of the minor axis is 2 * 13 = 26.
Therefore, the major axis length is 10 and the minor axis length is 26.
To find the x-intercepts and y-intercepts, we can set y = 0 and x = 0 respectively and solve for the corresponding values of x and y.
When y = 0:
x^2/25 + 0 = 1
x^2 = 25
x = ±5
So, the x-intercepts are (5, 0) and (-5, 0).
When x = 0:
0 + y^2/169 = 1
y^2/169 = 1
y^2 = 169
y = ±13
So, the y-intercepts are (0, 13) and (0, -13).
The foci of an ellipse can be found using the formula c = √(a^2 - b^2), where a and b are the lengths of the major and minor axes respectively.
In our case, a = 10 and b = 26, so we have:
c = √(10^2 - 26^2)
c = √(100 - 676)
c = √(-576)
Since we can't take the square root of a negative number in the real number system, it means that this ellipse has imaginary foci, and the center of the ellipse is the point (-12, 0).
Now, let's find the points of intersection of the ellipse with the line y = 1 - 2x.
Substitute y = 1 - 2x into the equation of the ellipse:
x^2/25 + (1 - 2x)^2/169 = 1
Simplifying this equation, we get:
(169x^2 + 338x - 7924)/338 = 0
Now solve this quadratic equation to find the x-values of the intersections. You can use the quadratic formula or other methods to solve it.