Find the differential equation of the circles which touch the x-axis at the origin.

Question 31 of 50

Q. Find the differential equation of the circles which touch the x-axis at the origin.

West Bengal WbchseWest Bengal HS (WBCHSE) Board 2019Subjective· 2mImportance★★★★★

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Circles touching the x-axis at the origin form the family $x^2+y^2=2ay$ ; eliminate the parameter $a$ by differentiating.

A circle touching the x-axis at the origin has its centre on the y-axis at $(0,a)$ with radius $|a|$ :

$x^2+(y-a)^2=a^2 \;\Rightarrow\; x^2+y^2-2ay=0 \;\Rightarrow\; x^2+y^2=2ay$

This is a one-parameter family (parameter $a$ ). Differentiate w.r.t. $x$ :

$2x+2yy'=2ay' \;\Rightarrow\; x+yy'=ay' \;\Rightarrow\; a=\dfrac{x+yy'}{y'}$

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Q.Find the differential equation of the circles which touch the x-axis at the origin.