Problem 13

The equation of a circle \(C_{1}\) is \(x^{2}+y^{2}=4\). The locus of the intersection of orthogonal tangents to the circle is the curve \(C_{2}\) and the locus of the intersection of perpendicular tangents to the curve \(C_{2}\) is the curve \(C_{3}\). Then : (a) \(C_{3}\) is a circle (b) The area enclosed by the curve \(C_{3}\) is \(8 \pi\) (c) \(C_{2}\) and \(C_{3}\) are circles with the same centre (d) none of these

Short Answer

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The correct answers are \(C_{3}\) is a circle (option a), \(C_{2}\) and \(C_{3}\) are circles with the same centre (option c). The area contained by \(C_{3}\) is not \(8\pi\), it is \(4\pi\), hence option b is incorrect, and option d is also incorrect.

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TABLE OF CONTENTS :

TABLE OF CONTENTS

Step 1: Identifying Characteristics of Curve \(C_{1}\)

From the given circle equation \(x^{2}+y^{2}=4\), we can identify that the center of the circle \(C_{1}\) is at the origin (0,0) and the radius is \(r=2\).

Step 2: Find the Locus for the Intersection Points of Orthogonal Tangents to \(C_{1}\)

It is known in geometry that the locus of the intersection points of perpendicular tangents to a given circle forms another circle with the same center. Therefore, \(C_{2}\) will be a circle with the same center as \(C_{1}\) which is (0, 0).

Step 3: Find the Locus for the Intersection Points of Orthogonal Tangents to \(C_{2}\)

Applying the same logic as in Step 2, the locus of the intersection points of the orthogonal tangents to the curve \(C_{2}\) will form another circle \(C_{3}\) with the same center (0,0).

Step 4: Compute the Area Covered by \(C_{3}\)

Since \(C_{3}\) has the same center as \(C_{1}\) and \(C_{2}\) and the same radius with \(C_{1}\), the area will be \(A = \pi r^{2} = \pi*2^{2} = 4\pi\).

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Chapter 1: Chapter 1

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Step 1: Identifying Characteristics of Curve \(C_{1}\)

Step 2: Find the Locus for the Intersection Points of Orthogonal Tangents to \(C_{1}\)

Step 3: Find the Locus for the Intersection Points of Orthogonal Tangents to \(C_{2}\)

Step 4: Compute the Area Covered by \(C_{3}\)

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More chapters from the book ‘Skills in Mathematics for All Engineering Entrance Examinations: Coordinate Geometry’

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Chapter 3

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