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  • If the chord of contact of the tangents from a point on the circle <img alt="\mathrm{x^2+y^2=a^2}" src="https://entrancecorner.oncodecogs.com/gif.latex?%5Cmathrm%7Bx%5E2&plus;y%5E2%3Da%5E2%7D"

If the chord of contact of the tangents from a point on the circle \mathrm{x^2+y^2=a^2} to the circle \mathrm{x^2+y^2=b^2} touch the circle \mathrm{x^2+y^2=c^2}, then the roots of the equation \mathrm{a x^2+2 b x+c=0} are necessarily
 

Option: 1

imaginary
 


Option: 2

 real and equal
 


Option: 3

real and unequal
 


Option: 4

rationa


Answers (1)

best_answer

 Let \mathrm{P(a \cos \theta, b \sin \theta)} be a point on \mathrm{x^2+y^2=a^2}
Chord of contact of tangents from P to circle 

\mathrm{x^2+y^2=b^2 \text{ is }x a \ \cos \theta+y a \ \sin \theta=b^2}
\mathrm{\Rightarrow \quad y=-\cot \theta x+\frac{b^2}{a} \operatorname{cosec} \theta \quad \quad \dots(i)}

Since line (i) is tangent to circle \mathrm{x^2+y^2=c^2}

\mathrm{\therefore\left(\frac{b^2}{a} \operatorname{cosec} \theta\right)^2=c^2\left(1+\cot ^2 \theta\right) \Rightarrow b^2=a c }

But \mathrm{D=4 b^2-4 a c=4 a c-4 a c=0}

\therefore Roots of equation \mathrm{a x^2+2 b x+c=0} are real and equal

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