Question

If a circle S(x,y)=0 touches the line x + y = 5 at the point (2, 3) and S(1, 2) = 0, then the radius of such circle is .

• A. $r=\frac{3}{2}\mathrm{units}$
• B. $r=\frac{1}{2}\mathrm{units}$
• C. $r=\frac{3}{\sqrt{2}}\mathrm{units}$
• D. $r=\frac{1}{\sqrt{2}}\mathrm{units}$

Answer 1

The correct option is D $r=\frac{1}{\sqrt{2}}\mathrm{units}$

$\mathrm{the}\mathrm{equation}\mathrm{of}\mathrm{family}\mathrm{of}\mathrm{circles}\mathrm{touching}\mathrm{the}\mathrm{line}x+y-5=0\mathrm{at}(2,3)\mathrm{is}(x-2{)}^2+(y-3{)}^2+\lambda (x+y-5)=0\mathrm{Also},(1,2)\mathrm{lies}\mathrm{on}\mathrm{it}\Rightarrow 1+1+\lambda (-2)=0\Rightarrow \lambda =1\mathrm{So},\mathrm{equation}\mathrm{of}\mathrm{required}\mathrm{circle}\mathrm{is}:x^2+y^2-3x-5y+8=0\mathrm{On}\mathrm{solving},\mathrm{we}\mathrm{get}:\mathrm{Radius}=\frac{1}{\sqrt{2}}\mathrm{units}$