The equation of the line of the shortest distance between the parabola y 2 = 4 x and the circle x 2 + y 2 − 4 x − 2 y + 4 = 0 is.

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TestHub · Accepted Answer

Line of shortest distance is normal for both parabola and circle Centre of circle is $(2, 1)$

Equation of normal to circle is $y - 1 = m (x - 2) \Rightarrow y = m x + (1 - 2 m)$

Equation of normal for a parabola is $y = m x - 2 am - a m^{3}$

Comparing (1) and (2)

$a m^{3} = - 1 \Rightarrow m^{3} = - 1 \Rightarrow m = - 1 (a = 1)$

Equation is $y - 1 = - x + 2 \Rightarrow x + y = 3$

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