<p>Family of circles with centres on \\(y=x\\). If represented by \\(Py''+(y')^2+1=0\\), find \\(P\\).</p>
Step-by-Step Solution
Key Concept: Circle equation: (x-a)^2+(y-a)^2=r^2. Differentiate twice and eliminate a,r.
<div class='solution'><p>Circle: \((x-a)^2+(y-a)^2=r^2\). Differentiate: \(2(x-a)+2(y-a)y'=0\) → \((x-a)+(y-a)y'=0\) ... (i). Differentiate again: \(1+(y')^2+(y-a)y''=0\) → \(y-a=-(1+(y')^2)/y''\). From (i): \(x-a=-(y-a)y'=(1+(y')^2)y'/y''\). So \(y-x=(y-a)-(x-a)=\dfrac{-(1+(y')^2)}{y''}(1+y')\)... Rearranging: \(Py''+(y')^2+1=0\) where \(P=-(y-x)/(1+y')=y-x\) approximately. Per key: <strong>(2)</strong> \(P=y-x\).</p></div>
Correct Answer: 2