question_answer

Consider the curve given by \[{{({{x}^{2}}+{{y}^{2}})}^{2}}=4({{x}^{2}}-{{y}^{2}})\].  Which     of the following is not true?

A)  The curve has two tangents parallel to X-axis

B)  The curve has two tangents parallel to Y-axis

C)  The area of the region bounded by this curve is less than 8

D)  All of the above

Correct Answer: D

Solution :

Given,  \[{{({{x}^{2}}+{{y}^{2}})}^{2}}=4({{x}^{2}}-{{y}^{2}})\] Graph of curve is Clearly, it has two tangent is parallel of X-axis and two tangent is parallel to Y-axis Polor coordinate of curve is \[{{r}^{2}}=4\,\cos \,2\theta \] Area of curve \[=\int_{-\pi /4}^{\pi /4}{{{r}^{2}}\,\,d\theta =4\,\,\int_{-\pi /4}^{\pi /4}{\cos \,2\theta \,\,d\theta }}\] \[=8\int_{0}^{\pi /4}{\cos \,2\theta \,\,d\theta =8\left[ \frac{\sin \,2\theta }{2} \right]_{0}^{\pi /4}=4}\] Hence, area of curve is less than 4. \[\therefore \] All of the options are true.