Hey bot , identify the type of conic section that has the equation 9x^2 +16y^2=144 and identify its domain and range
The given equation is a standard form equation of an ellipse. The center of the ellipse is at the origin, and the major axis is along the x-axis and the minor axis is along the y-axis.
The equation can be rewritten as:
(x^2)/4 + (y^2)/9 = 1
The domain of the ellipse is [-2, 2] (inclusive) along the x-axis, and the range of the ellipse is [-3, 3] (inclusive) along the y-axis.
To identify the type of conic section that has the equation 9x^2 + 16y^2 = 144, we need to analyze its coefficients.
First, let's compare the coefficients of x^2 and y^2. In this equation, the coefficient of x^2 is 9, and the coefficient of y^2 is 16.
Based on these coefficients, we can determine that the conic section is an ellipse.
Next, let's identify the domain and range of the given ellipse equation.
The domain refers to the possible values of x, while the range refers to the possible values of y.
Since the equation is 9x^2 + 16y^2 = 144, we can solve it for x or y to determine the domain and range.
Solving for x, we get: x^2 = (144 - 16y^2)/9.
To find the possible values of x, we need to consider the square root of both sides.
So, x = ± sqrt((144 - 16y^2)/9).
From this equation, we can see that as y varies from -∞ to +∞, the corresponding x values will be within the range of ± sqrt( (144 - 16y^2)/9 ).
Therefore, the domain is all real numbers (-∞, +∞), and the range is also all real numbers (-∞, +∞).