Question

Match the following:

Equations of the curve	Nature of the curve
A. $x=2(\cos \theta +\sin \theta )$ , $y=5(\cos \theta -\sin \theta )$	1. Parabola
B. $x=3(\cosh \theta +\sinh \theta )$ , $y=4(\cosh \theta -\sinh \theta )$	2. Ellipse
C. $x={\sin }^2\theta ,y=2\cos \theta$	3. Hyperbola
	4. Rectangular hyperbola

The correct match is :

• A. $A-1,B-2,C-3$
• B. $A-2,B-3,C-1$
• C. $A-3,B-2,C-1$
• D. $A-2,B-4,C-1$

Answer 1

The correct option is D $A-2,B-4,C-1$

$\left(A\right)x=2(\sin \theta +\cos \theta )$

$y=5(\cos \theta -\sin \theta )$

$\frac{5x+2y}{20}=\cos \theta$

$\frac{5x-2y}{20}=\sin \theta$

${\left(\frac{5x+2y}{20}\right)}^2+{\left(\frac{5x-2y}{20}\right)}^2=1\Rightarrow$ ellipse

$\left(B\right)\frac{x}{3}=\cos h\theta +\sin \theta ,\frac{y^2}{4}\cosh \theta -\sinh \theta$

$\left(\frac{4x+3y}{24}\right)=\cosh \theta ,\left(\frac{4x-3y}{24}\right)=\sinh \theta$

$\frac{x}{3}\times \frac{y}{4}=\frac{1}{4}$

$xy=3$ (rectangular hyperbola)

$\left(C\right)x={\sin }^2\theta$

$x=1-{\cos }^2\theta$

$x=1-(\frac{y}{2}{)}^2$

$y^2=4(1-x)\Rightarrow$ parabola