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  4. Let a circle C :( x - h )2+( y - k )2= r 2, k >0, touch the x-axis at (1,0). If the line x + y =0 intersects the circle C at P and Q such that the length of the chord P Q is 2 , then the value of h+k+r is equal to .

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Q. Let a circle $C :( x - h )^{2}+( y - k )^{2}= r ^{2}, k >0$, touch the $x$-axis at $(1,0)$. If the line $x + y =0$ intersects the circle $C$ at $P$ and $Q$ such that the length of the chord $P Q$ is $2$ , then the value of $h+k+r$ is equal to ______.

JEE MainJEE Main 2022Conic Sections

Solution:

$k = r$
$h =1$
$OP = r , PR =1$
$OR =\left|\frac{ r +1}{\sqrt{2}}\right|$
image
$r ^{2}=1+\frac{( r +1)^{2}}{2}$
$2 r ^{2}=2+ r ^{2}+1+2 r$
$r ^{2}-2 r -3=0$
$( r -3)( r +1)=0$
$r =3,-1$
$h + k + r =1+3+3$
$=7$