SOLUTION: Show that the equation (x^2)/(5-C) + (y^2)/ (9-C)=1 represents - an ellipse if C is any constant less than 5. - a hyperbola if C is any constant between 5 and 9. - no real loc

Algebra ->  Quadratic-relations-and-conic-sections -> SOLUTION: Show that the equation (x^2)/(5-C) + (y^2)/ (9-C)=1 represents - an ellipse if C is any constant less than 5. - a hyperbola if C is any constant between 5 and 9. - no real loc      Log On


   

Question 473981: Show that the equation (x^2)/(5-C) + (y^2)/ (9-C)=1 represents
- an ellipse if C is any constant less than 5.
- a hyperbola if C is any constant between 5 and 9.
- no real locus if C is greater than 9.
Answer by robertb(5830) About Me  (Show Source):
You can put this solution on YOUR website!
If C < 5, then ==> 5 - C >0, and 9 - C >0, hence the coefficients will be positive, and we get an equation for an ellipse.
If 5 < C < 9, then 0 > 5 - C > -4, while 4 > 9 - C > 0. This would give a hyperbola with the y-axis as transverse axis, and the x-axis as conjugate axis.
If C > 9, then both 5 - C and 9 - C are negative, and so the left side of the equation is negative while the right side is positive, a contradiction, hence no real locus is formed.