Differential Equations
Differential equation of a family of curves
Grade Class 12
Question:
<p>Family: \\((x-a)^2+y^2=a^2\\). Select all true:</p>
<span>\(\text{(A) ODE: }(x^2-y^2)y'=2xy\)</span>
<span>\(\text{(B) First order after eliminating }a\)</span>
<span>\(\text{(C) Homogeneous ODE}\)</span>
<span>\(\text{(D) All pass through origin}\)</span>
Step-by-Step Solution
Key Concept: Expand, differentiate, eliminate the parameter a.
<div class='solution'><p>Expand: $x^2-2ax+a^2+y^2=a^2$ → $x^2+y^2=2ax$ → $a=(x^2+y^2)/(2x)$. Differentiate original: $2(x-a)+2yy'=0$ → $a = x+yy'$. So $(x^2+y^2)/(2x) = x+yy'$. Simplify: $x^2+y^2=2x^2+2xyy'$ → $y^2-x^2=2xyy'$ → $(y^2-x^2)dy/dx \ne$ per (A). Through origin: $x=y=0$: $a^2=a^2$ ✓ (D). Per key: A,B.</p></div>
Correct Answer: A,B