JEE Advanced · Mathematics · 12. Circle

Tangents are drawn from the point \((17,7)\) to the circle \(x^2+y^2=169\).
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 circle is \(x^2+y^2=338\)

A Statement I is true, Statement II is true; Statement II is a correct explanation for Statement I
B Statement I is true, Statement II is true; Statement II is not a correct explanation for Statement I
C Statement I is true, Statement II is false
D Statement I is false, Statement II is true

Verified Solution

Answer & Solution

Correct Answer

(A) Statement I is true, Statement II is true; Statement II is a correct explanation for Statement I

Step-by-step Solution

Detailed explanation

Since, the tangents are perpendicular.
So, locus of perpendicular tangents to circle \(x^2+y^2=169\) is a director circle having equation
\[
x^2+y^2=338 .
\]

Apply the relevant identity from the chapter and substitute the values established in the previous step, keeping the original constraint in view.

Use the simplified expression to isolate the unknown, then plug the result back into the question setup to confirm the working is internally consistent.

Cross-check against a boundary or limiting case. Both the dimensional balance and the qualitative behaviour at the extreme should agree with the candidate answer.

Combine the steps above to identify the correct option. The remaining choices each fail at least one of the consistency, dimensional, or boundary checks.

Same subject

Column I	Column II
(A) In \(R ^2\), if the magnitude of the projection vector of the vector \(\alpha \hat{ i }+\beta \hat{ j }\) on \(\sqrt{3} \hat{ i }+\hat{ j }\) is \(\sqrt{3}\) and if \(\alpha=2+\sqrt{3} \beta\), then possible value (s) of \(\|\alpha\|\) is (are)	(P) 1
(B) Let a and b be real numbers such that the function \(f ( x )=\left\{\begin{array}{cc}-3 ax ^2-2, & x <1 \\ bx + a ^2, & x \geq 1\end{array}\right.\) is differentiable for all \(x \in R\). Then possible value ( s ) of a is (are)	(Q) 2
(C)Let \(\omega \neq 1\) be a complex cube roots of unity. If \(\left(3-3 \omega+2 \omega^2\right)^{4 n+3}\) \(+~\left(2+3 \omega-3 \omega^2\right)^{4 n+3}\) \(~+\left(-3+2 \omega+3 \omega^2\right)^{4 n+3}=0\) then possible value \((s)\) of \(n\) is (are)	(R) 3
(D) Let the harmonic mean of two positive real numbers \(a\) and \(b 4\). If \(q\) is a positive real number such that \(a , 5, q , b\) is an arithmetic progression, then the value \(( s )\) of \(\| q - a \|\) is (are)	(S) 4
	(T) 5

JEE Advanced 2015 Hard

A line with positive direction cosines passes through the point \(P(2,-1,2)\) and makes equal angles with the coordinate axes. The line meets the plane \(2 x+y+z=9\) at point \(Q\). The length of the line segment \(P Q\) equals

JEE Advanced 2009 Easy

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From JEE Advanced

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Tangents are drawn from the point \((17,7)\) to the circle \(x^2+y^2=169\).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 circle is \(x^2+y^2=338\)

Step-by-step Solution

Tangents are drawn from the point \((17,7)\) to the circle \(x^2+y^2=169\).
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 circle is \(x^2+y^2=338\)