QJEEAdv15Ma5121: JEE Advanced 2015 Mathematics - Class 12 JEEAdv15Ma5121 Solution: Consider the family of all circles whose centres lie on the…

Question

Verified Solution • 2025–26 Session moderate

Consider the family of all circles whose centres lie on the straight line

y=x,

If this family of circle is represented by the differential equation

Py'' + Qy' + 1 = 0,

where

P, Q

are functions of

x,y

and

y'

\left( {here\,\,\,y' = {{dy} \over {dx}},y'' = {{{d^2}y} \over {d{x^2}}}} \right)

then which of the following statements is (are) true?

Accepted Answer

Let equation of circle whose centre lie on straight line

y=x

be

(x-k)^2+(y-k)^2=r^2~~~~...(i)

Differentiating the above equation w.r.t.

x

, we get

\begin{aligned}&amp; 2(x-k)+2(y-k) y^{\prime}=0 \\\\\Rightarrow &amp; x+y y^{\prime}=k\left(1+y^{\prime}\right)~~~~~...(ii) \\\\\Rightarrow &amp; k=\frac{x+y y^{\prime}}{1+y^{\prime}}\end{aligned}

Differentiating the eq. (ii) w.r.t.

x

, we get

\begin{aligned}&amp; \Rightarrow 1+y y^{\prime \prime}+\left(y^{\prime}\right)^2=k y^{\prime \prime} \quad\left\{\because(u v)^{\prime}=u v^{\prime}+v u^{\prime}\right\} \\\\&amp; \Rightarrow 1+y y^{\prime \prime}+\left(y^{\prime}\right)^2=\left(\frac{x+y y^{\prime}}{1+y^{\prime}}\right) y^{\prime \prime} \\\\&amp; \Rightarrow 1+y y^{\prime \prime}+\left(y^{\prime}\right)^2+y^{\prime}+y y^{\prime} y^{\prime \prime}+\left(y^{\prime}\right)^3 \\\\&amp; =x y^{\prime \prime}+y y^{\prime} y^{\prime \prime} \\\\&amp; \Rightarrow y^{\prime \prime}(y-x)+\left(y^{\prime}\right)^2\left(1+y^{\prime}\right)+1+y^{\prime}=0 \\\\&amp; \Rightarrow y^{\prime \prime}(y-x)+y^{\prime}\left(y^{\prime}+\left(y^{\prime}\right)^2+1\right)+1=0\end{aligned}

Comparing the above equation with

p y^{\prime \prime}+\mathrm{Q} y^{\prime}

+1=0

, we get

\begin{aligned}&amp; P=y-x \text { and } \mathrm{Q}=y^{\prime}+\left(y^{\prime}\right)^2+1 \\\\\therefore &amp; P+Q=1-x+y+y^{\prime}+\left(y^{\prime}\right)^2\end{aligned}

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