identify the type of conic section that has the equation 16x^2+4y^2=16 and identify its domain and range
The given equation is 16x^2 + 4y^2 = 16, which can be simplified by dividing both sides by 16 to get x^2/1 + y^2/4 = 1. This is the equation of an ellipse centered at the origin with semi-major axis of length √4 = 2 and semi-minor axis of length √1 = 1.
Therefore, the type of conic section that has the given equation is an ellipse.
The domain of the ellipse is all real values of x that satisfy the equation, which is given by -2 ≤ x ≤ 2. The range of the ellipse is all real values of y that satisfy the equation, which is given by -1 ≤ y ≤ 1.
To identify the type of conic section that has the equation 16x^2 + 4y^2 = 16, we need to simplify the equation and compare it with the standard equations of conic sections.
Let's first divide the equation by 16 to obtain the simplified form:
x^2/1 + y^2/4 = 1
Now, we can compare this equation with the standard equations of conic sections:
1) If the equation is of the form x^2/a^2 + y^2/b^2 = 1, the conic section is an ellipse.
2) If the equation is of the form x^2/a^2 - y^2/b^2 = 1, the conic section is a hyperbola.
3) If the equation is of the form x^2/a^2 - y^2/b^2 = 0, the conic section is a parabola.
Since our equation is of the form x^2/1 + y^2/4 = 1, it matches the standard equation of an ellipse. Therefore, the given equation represents an ellipse.
Now, let's determine the domain and range of this ellipse:
The x-values of an ellipse can take any value, so the domain is (-∞, +∞).
For the y-values, we solve for y in terms of x:
4y^2 = 16 - 16x^2
y^2 = 4 - 4x^2
y = ±√(4 - 4x^2)
Here, the value under the square root cannot be negative, so:
4 - 4x^2 ≥ 0
-4x^2 ≥ -4
x^2 ≤ 1
-1 ≤ x ≤ 1
Hence, the range of the ellipse is [-∞, +∞] for y and [-1, 1] for x.
To summarize:
- The given equation represents an ellipse.
- The domain is (-∞, +∞).
- The range is [-∞, +∞] for y and [-1, 1] for x.