Find the differential equation of all circles passing through the origin and having their centers on the x-axis.

Question

Pohanginah · Accepted Answer

Given&amp;nbsp;Let the centre on x-axis be (h,0). In such case, h will also be the radius of the circle that passes through the origin.Finding the differential equation&amp;nbsp;h2+x2−2hx+y2+02−2(y)(0)=h2&amp;nbsp;x2−2hx+y2=h2−h2&amp;nbsp;x2+y2−2hx=0⋯(1)&amp;nbsp;Differentiating (1) w.r.t.x,&amp;nbsp;2x+2y&amp;nbsp;dy&amp;nbsp;&amp;nbsp;dx&amp;nbsp;−h=0&amp;nbsp;or&amp;nbsp;h=x+y&amp;nbsp;dy&amp;nbsp;&amp;nbsp;dx&amp;nbsp;⋯(2)&amp;nbsp;Putting the value of h from (2) in (1), we receivex2+y2−2x(x+y&amp;nbsp;dy&amp;nbsp;&amp;nbsp;dx&amp;nbsp;)=0&amp;nbsp;or&amp;nbsp;y2−x2−2xy&amp;nbsp;dy&amp;nbsp;&amp;nbsp;dx&amp;nbsp;=0&amp;nbsp;or&amp;nbsp;x2−y2+2xy&amp;nbsp;dy&amp;nbsp;&amp;nbsp;dx&amp;nbsp;=0

Jeffrey Jordon · Accepted Answer

Answer is given below (on video)

Find the differential equation of all circles passing through the

Answered question

Answer & Explanation

New Questions in High school geometry