Determine the type of curve represented by the equation [(x^2)/k + (y^2)/(k-16)]=1 in each of follow case: k<0
0<k<16
K>16
Mathas assajment
ellipse: x^2/a^2 + y^2/b^2
hyperbola: x^2/a^2 - y^2/b^2 = 1
So when do k and k-16 have the same or opposite sign?
Answer
Solutions
To determine the type of curve represented by the equation [(x^2)/k + (y^2)/(k-16)] = 1 for different values of k (<0, 0<k<16, and k>16), we need to look at the sign of the coefficients of x^2 and y^2.
First, let's simplify the equation to understand it better:
(x^2)/k + (y^2)/(k-16) = 1
1. When k < 0:
In this case, the coefficient of x^2 (1/k) is positive, but the coefficient of y^2 (1/(k-16)) is negative. It means that the x^2 term is positive, and the y^2 term is negative. This equation represents a type of curve known as an "ellipse." However, since the sign of the coefficient of the y^2 term is negative, the ellipse is vertically oriented.
2. When 0 < k < 16:
For this range of k values, both coefficients (1/k and 1/(k-16)) are positive. It means that both x^2 and y^2 terms are positive. Again, this equation represents an ellipse, but this time the ellipse is horizontally oriented.
3. When k > 16:
When k is greater than 16, both coefficients are positive, similar to the previous case. Therefore, it represents an ellipse, but this ellipse will be vertically oriented.
In summary:
- For k < 0, the equation represents a vertically oriented ellipse.
- For 0 < k < 16, the equation represents a horizontally oriented ellipse.
- For k > 16, the equation represents a vertically oriented ellipse.
These conclusions were determined by identifying the signs of the coefficients of the x^2 and y^2 terms in each case.