InterviewSolution
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InterviewSolution
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Find the equation of the circle passing through `(1,0)a n d(0,1)`and having the smallest possible radius.A. `x^(2)+y^(2)+x+y-2=0`B. `x^(2)+y^(2)=x+y`C. `x^(2)+y^(2)=1`D. none of these |
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Answer» Correct Answer - B Let the equation of the required circle be `x^(2)+y^(2)+2gx+2fy+c=0 " " ...(i)` This passes through A(1, 0) and B(0, 1). Therefore, `1+2g+c=0` and, `1+2f+c=0` `rArr g=-((c+1)/(2))` and, `f=-((c+1)/(2))` Let r be the radius of circle (i). Then, `r=sqrt(g^(2)+f^(2)-c)` `rArr=sqrt(((c+1)/(2))^(2)+((c+1)/(2))^(2)-c)` `rArr r=sqrt((c^(2)+1)/(2))rArr r^(2)=(1)/(2)(c^(2)+1)` Clearly , r is minimum when c=0 and the minimum value of r is `(1)/(sqrt(2))`. For c = 0, we have `g=-(1)/(2)` and `f=-(1)/(2)` Substituting the values of g, f and c in (i), we get `x^(2)+y^(2)-x-y=0` as the equation of the required circle. |
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