Question

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

Answer 1

Answer: (D)

All of the above

Answer 2

Given, ${{({{x}^{2}}+{{y}^{2}})}^{2}}=4({{x}^{2}}-{{y}^{2}})$ Graph of curve isClearly, 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.