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If the tangents are drawn to the circle \mathrm{x^2+y^2=12} at the points where it meets the circles \mathrm{x^2+y^2-5 x+3 y-2=0} then the greatest integer value of length of common chord is _____

Option: 1

6


Option: 2

1


Option: 3

4


Option: 4

5


Answers (1)

best_answer

We have circles \mathrm{\begin{aligned} S_1 & =x^2+y^2-12=0 \\ \text { and } S_2 & =x^2+y^2-5 x+3 y-2=0 \end{aligned}}-----(i) & (ii)

Circles are intersecting at points A and B.
Tangents are drawn to circle S1 = 0 at A and B, which meet
at point P(h, k).
AB is common chord of the circles, whose equation is

\mathrm{\begin{aligned} & S_1-S_2=0 . \\ & \text { or } 5 x-3 y-10=0 \end{aligned}} ----------(iii)

In the figure, \mathrm{C_1 M=\frac{|5(0)-3(0)-10|}{\sqrt{5^2+(-3)^2}}=\frac{10}{\sqrt{34}}}

In triangle \mathrm{C_1 M A, A M^2=C_1 A^2-C_1 M^2=12-\frac{100}{34}=\frac{154}{17}}

\mathrm{\therefore } Length of common chord

\mathrm{\begin{aligned} & A B=2 A M=2 \sqrt{\frac{154}{17}} \\ & \therefore \quad\left[2 \sqrt{\frac{154}{17}}\right]=6 \end{aligned} }

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vinayak

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