A variable circle is drawn passing through the origin 'O'. It int… - Vidyadip

MCQ

A variable circle is drawn passing through the origin $'O'.$ It intersects $X\, \& \,Y$ axis respectively at points $A\, \& \,B$ such that $OA + 2OB = K$ (non zero constant), then circle always passes through a fixed point $P$ other than origin. $P$ lies on -

A
the $x-$ axis
✓
the line $y = 2x$
C
the line $x = 2y$
D
the $y-$ axis

✓

Answer

Correct option: B.

the line $y = 2x$

b
Let circle is $x^{2}+y^{2}-2 \alpha x-2 \beta \gamma=0$

$\because \mathrm{OA}+2 \mathrm{OB}=\mathrm{K} \Rightarrow 2 \alpha+4 \beta=\mathrm{k}$

$\Rightarrow$ circle is $x^{2}+y^{2}-(k-4 \beta) x-2 \beta \gamma=0$

$\left(x^{2}+y^{2}-k x\right)+2 \beta(2 x-y)=0$

$\Rightarrow$ Fixed point lies on $2 \mathrm{x}=\mathrm{y}$

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