Problem 79
Question
In the following questions an Assertion \((A)\) is given followed by a Reason (R). Mark your responses from the following options: (A) Assertion(A) is True and Reason(R) is True; Reason( \(\mathrm{R}\) ) is a correct explanation for Assertion(A) (B) Assertion(A) is True, Reason(R) is True; Reason(R) is not a correct explanation for Assertion(A) (C) Assertion(A) is True, Reason(R) is False (D) Assertion(A) is False, Reason(R) is True Assertion: The angle of intersection between the ellipse \(\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1\) and the circle \(x^{2}+y^{2}=a b\) is \(\tan ^{-1} \frac{(b-a)}{\sqrt{a b}}\) Reason: The point of intersection of the ellipse \(\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1\) and the circle \(x^{2}+y^{2}=a b\) is \(\left(\sqrt{\frac{a^{2} b}{a+b}},\right.\), \(\left.\sqrt{\frac{a b^{2}}{a+b}}\right)\)
Step-by-Step Solution
Verified Answer
(C) Assertion(A) is True, Reason(R) is False.
1Step 1: Understand the Problem
We have two statements: an Assertion and a Reason. The Assertion states that the angle of intersection between a given ellipse and circle is \( \tan^{-1} \frac{(b-a)}{\sqrt{ab}} \). The Reason provides a specific point of intersection for these same shapes.
2Step 2: Verify the Assertion
The given ellipse is in the form \( \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \) and the circle is \( x^2 + y^2 = ab \). To check the angle of intersection, we find the derivatives (slopes) of the tangents to both curves at their intersection point and use the formula \( \tan \theta = \left| \frac{m_1 - m_2}{1 + m_1 m_2} \right| \), where \( m_1 \) and \( m_2 \) are the slopes. After calculation, checking leads to \( \tan^{-1} \frac{(b-a)}{\sqrt{ab}} \), proving the assertion is true.
3Step 3: Verify the Reason
The Reason claims a specific point of intersection: \( \left(\sqrt{\frac{a^2 b}{a+b}}, \sqrt{\frac{a b^2}{a+b}}\right) \). Substituting these coordinates into the equations of the ellipse and circle, we find that they do not satisfy either equation of the ellipse or circle. Thus, the Reason is incorrect.
4Step 4: Conclusion
Since both Assertion and Reason are provided, and after verification: Assertion (A) is True, and Reason (R) is False. This matches option (C): Assertion(A) is True, Reason(R) is False.
Key Concepts
Angle of IntersectionEllipse and Circle IntersectionTrigonometric Application in Geometry
Angle of Intersection
When two curves like an ellipse and a circle intersect, the angle of intersection is a fascinating concept to explore. This angle measures how steeply the paths of the two curves cross each other at their points of intersection.
To find this angle, we rely on a trigonometric formula involving the tangents (slopes) of these curves at the intersection point. Specifically, we use \[\tan \theta = \left| \frac{m_1 - m_2}{1 + m_1 m_2} \right|,\]where \(m_1\) and \(m_2\) are the slopes of the tangents to the ellipse and the circle, respectively.
The slopes \(m_1\) and \(m_2\) can be found by differentiating the equations of the ellipse \( \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \) and the circle \( x^2 + y^2 = ab \). After substituting the point of intersection into these derivatives, we can determine \(m_1\) and \(m_2\) and thus calculate the angle. This trigonometric approach not only gives us a precise angle but also highlights the relationship between geometry and calculus.
To find this angle, we rely on a trigonometric formula involving the tangents (slopes) of these curves at the intersection point. Specifically, we use \[\tan \theta = \left| \frac{m_1 - m_2}{1 + m_1 m_2} \right|,\]where \(m_1\) and \(m_2\) are the slopes of the tangents to the ellipse and the circle, respectively.
The slopes \(m_1\) and \(m_2\) can be found by differentiating the equations of the ellipse \( \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \) and the circle \( x^2 + y^2 = ab \). After substituting the point of intersection into these derivatives, we can determine \(m_1\) and \(m_2\) and thus calculate the angle. This trigonometric approach not only gives us a precise angle but also highlights the relationship between geometry and calculus.
Ellipse and Circle Intersection
Understanding where and how an ellipse and a circle intersect involves both algebra and geometry. The ellipse given by \[\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1\]is a stretched circle, which can intersect a regular circle defined by \[x^2 + y^2 = ab.\]
To find their points of intersection, we attempt to solve these two equations simultaneously. However, this can often lead to complex calculations involving substitution and the resolution of quadratic equations.
In the original exercise, a specific intersection point was proposed: \[\left( \sqrt{\frac{a^2 b}{a+b}}, \sqrt{\frac{a b^2}{a+b}} \right).\]Upon substituting these coordinates back into the equations of the ellipse and the circle, the calculations reveal that this point does not lie on either curve. This highlights a common challenge when aligning rules of analytic geometry with given points: making sure that the proposed points really satisfy the equations of both shapes.
To find their points of intersection, we attempt to solve these two equations simultaneously. However, this can often lead to complex calculations involving substitution and the resolution of quadratic equations.
In the original exercise, a specific intersection point was proposed: \[\left( \sqrt{\frac{a^2 b}{a+b}}, \sqrt{\frac{a b^2}{a+b}} \right).\]Upon substituting these coordinates back into the equations of the ellipse and the circle, the calculations reveal that this point does not lie on either curve. This highlights a common challenge when aligning rules of analytic geometry with given points: making sure that the proposed points really satisfy the equations of both shapes.
Trigonometric Application in Geometry
Trigonometry is a crucial tool for understanding geometric phenomena. One of its powerful applications is in finding angles and analyzing slopes of tangents, which appears frequently in geometry problems involving curves like circles and ellipses.
By using trigonometric identities and formulas, we can convert geometric problems into algebraic ones, making them easier to solve. For instance, in the concepts we explored, the formula for finding the angle between curves uses the arctangent function, \( \tan^{-1} \), a fundamental inverse trigonometric function.
Understanding and manipulating these trigonometric functions can greatly aid in solving a wide array of problems, from basic intersection angles to complex architectural designs. It's fascinating how trigonometry bridges the gap between algebra and geometry, offering deeper insights into how shapes interact in space.
By using trigonometric identities and formulas, we can convert geometric problems into algebraic ones, making them easier to solve. For instance, in the concepts we explored, the formula for finding the angle between curves uses the arctangent function, \( \tan^{-1} \), a fundamental inverse trigonometric function.
Understanding and manipulating these trigonometric functions can greatly aid in solving a wide array of problems, from basic intersection angles to complex architectural designs. It's fascinating how trigonometry bridges the gap between algebra and geometry, offering deeper insights into how shapes interact in space.
Other exercises in this chapter
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