Question:
Consider the curve given by $ {{({{x}^{2}}+{{y}^{2}})}^{2}}=4({{x}^{2}}-{{y}^{2}}) $ . Which of the following is not true?
Consider the curve given by $ {{({{x}^{2}}+{{y}^{2}})}^{2}}=4({{x}^{2}}-{{y}^{2}}) $ . Which of the following is not true?
Updated On: Jun 23, 2024
- The curve has two tangents parallel to X-axis
- The curve has two tangents parallel to Y-axis
- The area of the region bounded by this curve is less than 8
- All of the above
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The Correct Option is D
Solution and Explanation
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.
$ {{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.
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Concepts Used:
Plane
A surface comprising all the straight lines that join any two points lying on it is called a plane in geometry. A plane is defined through any of the following uniquely:
- Using three non-collinear points
- Using a point and a line not on that line
- Using two distinct intersecting lines
- Using two separate parallel lines
Properties of a Plane:
- In a three-dimensional space, if there are two different planes than they are either parallel to each other or intersecting in a line.
- A line could be parallel to a plane, intersects the plane at a single point or is existing in the plane.
- If there are two different lines that are perpendicular to the same plane then they must be parallel to each other.
- If there are two separate planes which are perpendicular to the same line then they must be parallel to each other.