KCET · Maths · Differential Equations
If \(\cos ^{-1}\left(\frac{y}{b}\right)=n \log \left(\frac{x}{n}\right)\), then
- A \(x y_{1}=n \sqrt{b^{2}-y^{2}}\)
- B \(x y_{1}+n \sqrt{b^{2}-y^{2}}=0\)
- C \(y_{1}=x \sqrt{b^{2}-y^{2}}\)
- D \(x y_{1}-\sqrt{b^{2}-y^{2}}=0\)
Answer & Solution
Correct Answer
(B) \(x y_{1}+n \sqrt{b^{2}-y^{2}}=0\)
Step-by-step Solution
Detailed explanation
Given, \(\cos ^{-1}\left(\frac{y}{b}\right)=n \log \left(\frac{x}{n}\right) \quad\left(\because y_{1}=\frac{d y}{d x}\right)\)
Differentiating w.r.t. ' \(x\) '
\[
\begin{gathered}
-\frac{1}{\sqrt{1-(y / b)^{2}}} \cdot \frac{y_{1}}{b}=n \cdot \frac{1}{(x / n)} \cdot \frac{1}{n} \\
-\frac{b}{\sqrt{b^{2}-y^{2}}} \cdot \frac{y_{1}}{b}=\frac{n^{2}}{x} \cdot \frac{1}{n} \\
\Rightarrow \quad-\frac{y_{1}}{\sqrt{b^{2}-y^{2}}}=\frac{n}{x} \\
\Rightarrow \quad-x y_{1}=n \sqrt{b^{2}-y^{2}} \\
\Rightarrow \quad x y_{1}+n \sqrt{b^{2}-y^{2}}=0
\end{gathered}
\]
Differentiating w.r.t. ' \(x\) '
\[
\begin{gathered}
-\frac{1}{\sqrt{1-(y / b)^{2}}} \cdot \frac{y_{1}}{b}=n \cdot \frac{1}{(x / n)} \cdot \frac{1}{n} \\
-\frac{b}{\sqrt{b^{2}-y^{2}}} \cdot \frac{y_{1}}{b}=\frac{n^{2}}{x} \cdot \frac{1}{n} \\
\Rightarrow \quad-\frac{y_{1}}{\sqrt{b^{2}-y^{2}}}=\frac{n}{x} \\
\Rightarrow \quad-x y_{1}=n \sqrt{b^{2}-y^{2}} \\
\Rightarrow \quad x y_{1}+n \sqrt{b^{2}-y^{2}}=0
\end{gathered}
\]
See the Complete Solution
Get step-by-step explanations for this and 2.5 Lakh+ more JEE, NEET & CET questions.
- Unlock all solutions
- Practice the full chapter
- Track accuracy across PYQs
4.8 rated on Google Play · 14,000+ reviews
More questions from Maths
- A space vector makes the angles \(150^{\circ}\) and \(60^{\circ}\) with the positive direction of \(x\)-and \(y\)-axes. The angle made by the vector with the positive direction \(z\)-axis isKCET 2010 Easy
- If \(\mathbf{a}\) is a vector perpendicular to both \(\mathbf{b}\) and \(\mathbf{c}\), thenKCET 2013 Easy
- If \(\omega\) is an imaginary cube root of unity, then the value of \(\left(1-\omega+\omega^{2}\right) \cdot\left(1-\omega^{2}+\omega^{4}\right) \cdot\left(1-\omega^{4}+\omega^{8}\right) \cdot \ldots\)
(2n factors) isKCET 2011 Medium - If \( \left(\frac{1-i}{1+i}\right)^{96}=a+i b \) then \( (a, b) \) isKCET 2018 Medium
- The coordinates of the centre of the smallest circle passing through the origin and having \(\mathrm{y}=\mathrm{x}+1\) as a diameter areKCET 2009 Medium
- If \( x=a \cos ^{3} \theta, y=a \sin ^{3} \theta \), then \( 1+\left(\frac{d y}{d x}\right)^{2} \) isKCET 2015 Easy
More PYQs from KCET
- Read the statements A and B and select the correct option.
Statement A: Atherosclerosis is a disease characterized by the thickening of arterial walls.
Statement B: Deposition of cholesterol and triglycerides in the arterial walls causes atherosclerosis.KCET 2005 Easy - If \( f(x)=8 x^{2}, g(x)=x^{\frac{1}{3}} \) then \( f o g(x) \) isKCET 2017 Hard
- Two thin plano-convex lenses each of focal length \(f\) are placed as shown in the figure. The ratio of their effective focal lengths in the three cases is
KCET 2012 Medium - A moving coil galvanometer is converted into an ammeter of range 0 to \(5 \mathrm{~mA}\). The galvanometer resistance is \(90 \Omega\) and the shunt resistance has a value of \(10 \Omega\). If there are \(\mathbf{5 0}\) divisions in the galvanometer-turned-ammeter on either sides of zero, its current sensitivity isKCET 2023 Medium
- Let \(f: R \rightarrow R\) be defined by \(f(x)=x^2+1\). Then, the pre images of 17 and -3 . respectively areKCET 2024 Easy
- Which of the following sentences is correct?KCET 2015 Medium