Question

If the radius of two concentric circles are the roots of the equation $x^2-16x+63=0$ and the centre lies on the line $y=x+5$. Then the equation of circles are:

• A. $x^2+y^2-8x-18y+16=0$ and $x^2+y^2-8x-18y+49=0$
• B. $x^2+y^2-8x-18y+16=0$ and $x^2+y^2-8x-18y+48=0$
• C. $x^2+y^2-8x-18y+81=0$ and $x^2+y^2-8x-18y+45=0$
• D. None of these

Answer 1

The correct option is B $x^2+y^2-8x-18y+16=0$ and $x^2+y^2-8x-18y+48=0$

$\Rightarrow x^2-16x+63=0\Rightarrow x^2-9x-7x+63=0\Rightarrow x(x-9)-7(x-9)=0\Rightarrow x=7,9.\therefore Radius=7,9.Letcentrebe(h,k)\Rightarrow k=h+5Eq.ofcircle1:\Rightarrow (x-h{)}^2+(y-k{)}^2=49Equationofcircle2:(x-h{)}^2+(y-k{)}^2=81.AlloptionsA,B\&Csatisfycentrecoordinatesk=h+5,\because k=9,h=4.Radius=\sqrt{g^2+f^2-c.}InB)option,theabovevalueis7forIcircle\&9forIIcircle(\therefore Biscorrect).$