The curve represented by the equations

x = 3(cost + sint)

y = 4(cost - sint) is

Concept:

1. Equation of circle x2 + y2 = r2

2. Equation of an ellipse \(\frac{{{x^2}}}{{{a^2}}}\; + \frac{{{y^2}}}{{{b^2}}} = 1\)

3. Equation of Parabola y2 = 4ax, x2 = 4ay

4. Equation of hyperbola \(\frac{{{x^2}}}{{{a^2}}} - \;\frac{{{y^2}}}{{{b^2}}} = 1\)

5.If a2 = b2 then hyperbola is called rectangular hyperbola and x2 − y2 = a2 is the general form of a rectangular hyperbola

Calculation:

Given:

x = 3(cost + sint)

y = 4(cost - sint) 

\({\left( {\frac{x}{3}} \right)^2} = 1 + sin2t\) ----(1)

\({\left( {\frac{y}{4}} \right)^2} = 1 - sin2t\) ----(2)

Adding r=equation 1 and 2;

\(\frac{{{x^2}}}{9} + \frac{{{y^2}}}{{16}} = 2\)  

Hence, the given curve represents an ellipse.