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.You need to login to perform this action.
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