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Forming Differential equations
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Forming Differential equations
Last updated at May 29, 2023 by Teachoo
Question 6 Form the differential equation of the family of circle touching the π¦βππ₯ππ at origin. General Equation of Circle (π₯βπ)^2+(π¦βπ)^2=π^2 where Centre at (π , π) and Radius is r If circle touches y-axis at origin, Center will be at x-axis So, Center = (a, 0) & Radius = a Thus, equation of circle becomes (π₯βπ)^2+(π¦β0)^2=π^2 (π₯βπ)^2+π¦^2=π^2 π₯^2+π^2β2ππ₯+π¦^2=π^2 π₯^2β2ππ₯+π¦^2=π^2βπ^2 π₯^2β2ππ₯+π¦^2=0 2ππ₯=π₯^2+π¦^2 Differentiating Both Sides w.r.t. π₯ (π(2ππ₯))/ππ₯=π(π₯^2 )/ππ₯+π(π¦^2 )/ππ₯ 2a = 2x + 2y ππ¦/ππ₯ a = x + yyβ β¦(1) β¦(2) From (1) 2ππ₯=π₯^2+π¦^2 Putting value of a from (2) 2π₯(π₯+π¦π¦^β²)=π₯^2+π¦^2 2π₯^2+2π₯π¦π¦^β²=π₯^2+π¦^2 2π₯^2βπ₯^2+2π₯π¦π¦^β²=+π¦^2 ππππ^β²+π^π=π^π is the required differential equation.