Definition of a circle

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Question

$A circle touches the line 4 x + 3 y = 12 at D and positive x, y axes. Let DE be$
$diameter of circle and OE intersect the line at F, then [4 DF] is equal to ([.] is GIF)$ $(Provided OE < OD and O being origin) .$

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Solution

$Equation of circle is x^{2} + y^{2} - 2 x - 2 y + 1 = 0 D is (\frac{9}{5}, \frac{8}{5})$
$taking (1, 1) as and point E is (\frac{1}{5}, \frac{2}{5})$
$OE has equation y = 2 x$
$\Rightarrow F is (\frac{6}{5}, \frac{12}{5}) \Rightarrow DF = 1$

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