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  • The length of the intercept cut off from the line L: y=x  on Circle <img alt="(x-1)^2+(y-1)^2=1" src="https://entrancecorner.oncodecogs.com/gif.latex?%28x-1%29%5E2&plus;%28y-1%29%5E2%

The length of the intercept cut off from the line L: y=x  on Circle (x-1)^2+(y-1)^2=1 is ?

Option: 1

0


Option: 2

1


Option: 3

1.5


Option: 4

2


Answers (1)

best_answer

 

 

Intercepts Made by Circle on the Axis -

Intercepts Made by Circle on the Axis 
The equation of Circle is :

\mathrm{x^2+y^2+2gx+2fy+c=0}

Length of x-intercept :\mathrm{2\sqrt{g^2-c}}

Length of y-intercept : \mathrm{2\sqrt{f^2-c}}

 

\begin{array}{l}{\text { from the figure }} \\ {\text { length of intercepts on } X-\text { axis and } Y-\text { axis are }|A B| \text { and }|C D|} \\ {|A B|=\left|x_{2}-x_{1}\right|,|C D|=\left|y_{2}-y_{1}\right|} \\ {y=0, \text { circle intersects the } X-\text { axis }} \\ {\Rightarrow x^{2}+2 g x+c=0} \\ {\text { since, circle intersects } X-\text { axis at two points } A\left(x_{1}, 0\right) \text { and } B\left(x_{2}, 0\right)} \\ {\text { then, } x_{1}+x_{2}=-2 g x, x_{1} x_{2}=c} \\ {|A B|=\left|x_{2}-x_{1}\right|=\sqrt{\left(x_{2}+x_{1}\right)^{2}-4 x_{1} x_{2}}} \\ {\text { similarly, }} \\ {|C D|=2 \sqrt{f^{2}-c}}\end{array}

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(x-1)^2+(y-1)^2=1\\ \text{circle with centre (1,1) and radius r=1}\\ \text{line y=x is also pass through (1,1) so length of intercept = diameter of circle=2}\\

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