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  • The angle between the tangents drawn from the origin to the circle  <img alt="\mathrm{(x-7)^2+(y+1)^2=25}" src="https://entrancecorner.oncodecogs.com/gif.latex?%5Cmathrm%7B%28x-7%29%5E2a

The angle between the tangents drawn from the origin to the circle  \mathrm{(x-7)^2+(y+1)^2=25}is
 

Option: 1

\frac{\pi}{3}


Option: 2

\frac{\pi}{6}


Option: 3

{\frac{\pi}{2}}


Option: 4

\frac{\pi}{8}


Answers (1)

best_answer

\begin{aligned} &\mathrm{S}:(\mathrm{x}-7)^2+(\mathrm{y}+1)^2=5^2, \mathrm{C}(7,-1), \mathrm{r}=5\\ &\text{or, }\mathrm{S=x^2+y^2-14 x+2 y+25=0.}\\ &\text{Here }\left(\mathrm{x}_1, \mathrm{y}_1\right)=(0,0) \therefore \mathrm{S}_1=0+25=25\\ &\mathrm{T}=\mathrm{xx}_1+\mathrm{yy}_1-7\left(\mathrm{x}+\mathrm{x}_1\right)+\left(\mathrm{y}+\mathrm{y}_1\right)+25\\ &\text{Put } \mathrm{x}_1=\mathrm{y}_1=0\\ &\mathrm{T}=-7 \mathrm{x}+\mathrm{y}+25. \\ \end{aligned}

Pair of tangents through origin to circle is

\begin{aligned} &\mathrm{T}^2=\mathrm{SS}_1 \Rightarrow(-7 \mathrm{x}+\mathrm{y}+25)^2=\left(\mathrm{x}^2+\mathrm{y}^2-14 \mathrm{x}+2 \mathrm{y}+25\right)(25)\\ &\mathrm{a}+\mathrm{b}= \text{coefficient of }\mathrm{x}^2+ \text{coefficient of }\mathrm{y}^2=(49+1)-(25+25)=0\\ &\therefore \text{Angle between tangents =}\pi / 2 \end{aligned}
 

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chirag

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