TS EAMCET · Maths · Differential Equations

Statement I : The differential equation corresponding to the family of circles having their centres on Y-axis and fixed radius \(k\) is \(\left(x^2-k^2\right)\left(\frac{d y}{d x}\right)^2+x^2=0\) Statement II : The differential equation of corresponding to the family of circles passing through the origin and having their centres on \(\mathrm{X}\)-axis is \(x^2-y^2+2 x y \frac{d y}{d x}=0\) Which of the above statements is (are) true?

A Statement I is true, but Statement II is false
B Statement II is true, but Statement I is false
C Both Statement I and Statement II are true
D Both Statement I and Statement II are false

Verified Solution

Answer & Solution

Correct Answer

(C) Both Statement I and Statement II are true

Step-by-step Solution

Detailed explanation

Statement I:- Let center \({C}=(0, \alpha)\) and radius \(={k}\) \(\Rightarrow\) equation of circle is \(\begin{array}{l} x ^2+( y -\alpha)^2= k ^2 ...(i)\\ \Rightarrow \alpha= x \frac{ dx }{ dy }+ y ...(ii) \end{array}\) Putting value of \(\alpha\) in equation (i)…

Apply the relevant identity from the chapter and substitute the values established in the previous step, keeping the original constraint in view.

Use the simplified expression to isolate the unknown, then plug the result back into the question setup to confirm the working is internally consistent.

Cross-check against a boundary or limiting case. Both the dimensional balance and the qualitative behaviour at the extreme should agree with the candidate answer.

Combine the steps above to identify the correct option. The remaining choices each fail at least one of the consistency, dimensional, or boundary checks.

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