Question:
The solution of
\[
(1+y^2)\,dx-xy\,dy=0,\quad y(1)=0
\]
represents a conic. Its eccentricity is
The solution of
\[
(1+y^2)\,dx-xy\,dy=0,\quad y(1)=0
\]
represents a conic. Its eccentricity is
Show Hint
For the hyperbola
\[
\frac{x^2}{a^2}-\frac{y^2}{b^2}=1,
\]
eccentricity is
\[
e=\sqrt{1+\frac{b^2}{a^2}}.
\]
Updated On: Jun 25, 2026
- \(2\)
- \(\dfrac{1}{e}\)
- \(1\)
- \(\sqrt{2}\)
Show Solution
Verified By Collegedunia
The Correct Option is D
Solution and Explanation
Step 1: Rearrange the differential equation.
Given: \[ (1+y^2)\,dx-xy\,dy=0 \] Rearranging, \[ (1+y^2)\,dx=xy\,dy \] Therefore, \[ \frac{dx}{x}=\frac{y}{1+y^2}\,dy \]
Step 2: Integrate both sides.
Integrating, \[ \int \frac{1}{x}\,dx = \int \frac{y}{1+y^2}\,dy \] \[ \log x = \frac12\log(1+y^2)+C \] Multiply by \(2\): \[ 2\log x=\log(1+y^2)+C \] Using logarithm properties: \[ \log x^2=\log(1+y^2)+C \] Hence, \[ x^2=C(1+y^2) \]
Step 3: Use the initial condition.
Given: \[ y(1)=0 \] Substitute: \[ 1=C(1+0) \] \[ C=1 \] Therefore, \[ x^2=1+y^2 \] or \[ x^2-y^2=1 \]
Step 4: Identify the conic.
The equation \[ x^2-y^2=1 \] represents a hyperbola.
Comparing with \[ \frac{x^2}{a^2}-\frac{y^2}{b^2}=1, \] we get \[ a^2=1,\quad b^2=1 \] Eccentricity of hyperbola is \[ e=\sqrt{1+\frac{b^2}{a^2}} \] So, \[ e=\sqrt{1+\frac11} \] \[ e=\sqrt{2} \]
Step 5: Final conclusion.
Therefore, the eccentricity is \[ \boxed{\sqrt{2}} \]
Given: \[ (1+y^2)\,dx-xy\,dy=0 \] Rearranging, \[ (1+y^2)\,dx=xy\,dy \] Therefore, \[ \frac{dx}{x}=\frac{y}{1+y^2}\,dy \]
Step 2: Integrate both sides.
Integrating, \[ \int \frac{1}{x}\,dx = \int \frac{y}{1+y^2}\,dy \] \[ \log x = \frac12\log(1+y^2)+C \] Multiply by \(2\): \[ 2\log x=\log(1+y^2)+C \] Using logarithm properties: \[ \log x^2=\log(1+y^2)+C \] Hence, \[ x^2=C(1+y^2) \]
Step 3: Use the initial condition.
Given: \[ y(1)=0 \] Substitute: \[ 1=C(1+0) \] \[ C=1 \] Therefore, \[ x^2=1+y^2 \] or \[ x^2-y^2=1 \]
Step 4: Identify the conic.
The equation \[ x^2-y^2=1 \] represents a hyperbola.
Comparing with \[ \frac{x^2}{a^2}-\frac{y^2}{b^2}=1, \] we get \[ a^2=1,\quad b^2=1 \] Eccentricity of hyperbola is \[ e=\sqrt{1+\frac{b^2}{a^2}} \] So, \[ e=\sqrt{1+\frac11} \] \[ e=\sqrt{2} \]
Step 5: Final conclusion.
Therefore, the eccentricity is \[ \boxed{\sqrt{2}} \]
Was this answer helpful?
0
0
Top AP EAPCET Mathematics Questions
- $D = \left\{x\in\mathbb{R}: f\left(x\right) =\sqrt{\frac{x - \left|x\right|}{x - \left[x\right]}} \text{is defined} \right\}$ and $C$ be the range of the real function $g(x) = \frac{2x}{4 + x^{2}}$. Then $D \cap C$
- $f\left(x\right) = \frac{x}{e^{x}-1} + \frac{x}{2} + 2 \cos^{3} \frac{x}{2} $ on $R-\left\{0\right\} $ is
- Consider the following lists.
List-I List-II (A) $f(x) = \frac{|x+2|}{x+2} , x \ne -2 $ (I) $[\frac{1}{3} , 1 ]$ (B) $(x)=|[x]|,x \in [R$ (II) Z (C) $h(x) = |x - [x]| , x \in [R$ (III) W (D) $f(x) = \frac{1}{2 - \sin 3x} , x \in [R$ (IV) [0, 1) (V) { -1, 1} - If $[x]$ is the greatest integer less than or equal to $x$ and $|x|$ is the modulus of $x$. then the system of three equations $2x + 3 | y | + 5[z] = 0, x + |y| - 2[z] = 4, x + |y| + |z| = 1$ has
- Investigate the values of $\lambda$ and $\mu$ for the system $x+2 y+3 z=6, x+3 y+5 z=9$, $2 x+5 y+\lambda z=\mu$ and match the values in List - I with the items in List - II.
List I List II (A) $\lambda=8, \mu \neq 15$ 1. Infinitely many solutions (B) $\lambda \neq 8, \mu \in R$ 2. No solution (C) $\lambda=8, \mu=15$ 3. Unique solution
View More Questions
Top AP EAPCET Differential Equations Questions
- If a curve passes through (1, 2) and has the slope of its tangent \(1 - \frac{1}{x^2}\) at a point \((x, y)\), then the equation of that curve is
- The differential equation having \( y = (a + b)e^{cx+d} \) as its general solution, where \( a, b, c, d \) are arbitrary constants, is
- The integrating factor of the linear differential equation \( \frac{dy}{dx} = \frac{1}{4x + 3y} \) is
- If the solution of \( \frac{dy}{dx} = y \log_e 0.5 = 0 \), \( y(0) = 1 \), and \( y(x) \to k \) as \( x \to \infty \), then \( k = \)
- The function \( y = Ae^x + Be^{-2x} \) satisfies which of the following differential equations?
View More Questions
Top AP EAPCET Questions
- A body is projected from the ground at an angle of $\tan^{-1} \left( \frac{8}{7}\right)$ with the horizontal. The ratio of the maximum height attained by it to its range is
- The four quantum numbers of the valence electron of potassium are :
- A lens forms real and virtual images of an object, when the object is at $u_{1}$ and $u_{2}$ distances respectively. If the size of the virtual image is double that of the real image, then the focal length of the lens is (take, the magnification of the real image as $m$ )
- Two spheres $P$ and $Q$, each of mass $200\, g$ are attached to a string of length one metre as shown in the figure. The string and the spheres are then whirled in a horizontal circle about $O$ at a constant angular speed. The ratio of the tension in the string between $P$ and $Q$ to that of between $P$ and $O$ is $(P$ is at mid-point of the line joining $O$ and $Q$ )
- The volume of $0.02\, M$ acidified permanganate solution required for complete reaction of $60\, mL$ of $0.01\, MI^{-}$ ion solution to form $I_2$ in $mL$ is
View More Questions