Consider the family of all circles whose centres lie on the straight line $y=x$. If this family of c

Question

Consider the family of all circles whose centres lie on the straight line $y=x$. If this family of circles is represented by the differential equation $P y^{\prime \prime}+Q y^{\prime}+1=0$, where $P$, $Q$ are functions of $x, y$ and $y^{\prime}$ (here $y^{\prime}=\frac{d y}{d x}, y^{\prime \prime}=\frac{d^{2} y}{d x^{2}}$ ), then which of the following statements is (are) true?\

Professor PC Thomas Classes · Answer

(b, c) : $(x-\alpha)^{2}+(y-\alpha)^{2}=r^{2}$\ $\Rightarrow x^{2}+y^{2}-2 \alpha x-2 \alpha y+2 \alpha^{2}-r^{2}=0$\ Differentiating (i) w.r.t. $x$, we get\ $2 x+2 y y^{\prime}-2 \alpha-2 \alpha y^{\prime}=0$\ $	herefore \quad \alpha=\frac{x+y y^{\prime}}{1+y^{\prime}}$\ Differentiating (ii), we get\ $1+y^{\prime 2}+y y^{\prime \prime}-\alpha y^{\prime \prime}=0$\ $\Rightarrow \quad 1+y^{\prime 2}+y y^{\prime \prime}-\frac{x+y y^{\prime}}{\left(1+y^{\prime}ight)} y^{\prime \prime}=0$\ [From (iii)]\ Collecting terms, we get $$ (y-x) y^{\prime \prime}+\left(1+y^{\prime}+y^{\prime 2}ight) y^{\prime}+1=0 $$ Now compare with given differential equation.\