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Question

Tangents are drawn from the point 17,7 to the circle x2+y2=169

Statement I: The tangents are mutually perpendicular.

Statement II: The locus of the points from which mutually perpendicular tangents can be drawn to the given circles is x2+y2=338


A

Statement I is correct, Statement II is correct; Statement II is the correct explanation for Statement I

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B

Statement I is correct, Statement II is correct; Statement II is not a correct explanation for Statement I

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C

Statement I is correct, Statement II is correct

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D

Statement I is incorrect, Statement II is correct

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Solution

The correct option is A

Statement I is correct, Statement II is correct; Statement II is the correct explanation for Statement I


Explanation for the correct option:

Statement I:

Calculating distance CB

We know that distance between 2 points x1,y1and x2,y2is x2-x12+y2-y12

CB=172+72=338

Finding the equation of the director circle:

The equation of the director circle of the circle x2+y2=a2 is x2+y2=2a2
And we also know that the locus of the point of intersection of two perpendicular tangents to a given circle is known as its director circle.

The equation of the director circle in this case is x2+y2=2132=338The point 17,7 lies on the director circle since it satisfies the equation of the director circle.Therefore by definition, the tangents are mutually perpendicular

Hence statement I is correct

Statement II:

By definition of the director circle: the locus of the points from which mutually perpendicular tangents can be drawn to the given circles is its director circle

And here, the equation of the director circle is x2+y2=338

Therefore statement II is correct and it explains statement I

Hence, option (A) is the correct answer.


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