Find the angle to intersection of the following curves :
$2 y^{2}=x^{3}$ and $y^{2}=32 x$
Given:
Curves $2 y^{2}=x^{3} \ldots(1)$
$\& y^{2}=32 x$ .....(2)
First curve is $2 y^{2}=x^{3}$
Differentiating above w.r.t $x$,
$\Rightarrow 4 y \cdot \frac{d y}{d x}=3 x^{2}$
$\Rightarrow m_{1}=\frac{d y}{d x}=\frac{3 x^{2}}{4 y} \ldots(3)$
Second curve is $y^{2}=32 x$
$\Rightarrow 2 y \cdot \frac{d y}{d x}=32$
$\Rightarrow y \cdot \frac{d y}{d x}=16$
$\Rightarrow m_{2}=\frac{d y}{d x}=\frac{16}{y} \ldots(4)$
Substituting (2) in (1), we get
$\Rightarrow 2 y^{2}=x^{3}$
$\Rightarrow 2(32 x)=x^{3}$
$\Rightarrow 64 x=x^{3}$
$\Rightarrow x^{3}-64 x=0$
$\Rightarrow x\left(x^{2}-64\right)=0$
$\Rightarrow x=0 \&\left(x^{2}-64\right)=0$
$\Rightarrow x=0 \& \pm 8$
Substituting $x=0 \& x=\pm 8$ in (1) in (2),
$y^{2}=32 x$
when $x=0, y=0$
when $x=8$
$\Rightarrow y^{2}=32 \times 8$
$\Rightarrow y^{2}=256$
$\Rightarrow y=\pm 16$
Substituting above values for $m_{1} \& m_{2}$, we get,
when $x=0, y=16$
$m_{1}=\frac{d y}{d x}$
$\Rightarrow \frac{3 \times 0^{2}}{4 \times 8}=0$
when $x=8, y=16$
$m_{1}=\frac{d y}{d x}$
$\Rightarrow \frac{3 \times 8^{2}}{4 \times 16}=3$
Values of $m_{1}$ is $0 \& 3$
when $x=0, y=0$,
$m_{2}=\frac{d y}{d x}$
$\Rightarrow \frac{16}{y}=\frac{16}{0}=\infty$
when $y=16$
$m_{2}=\frac{d y}{d x}$
$\Rightarrow \frac{16}{y}=\frac{16}{16}=1$
Values of $m_{2}$ is $\infty \& 1$
when $m_{1}=0 \& m_{2}=\infty$
$\Rightarrow \tan \theta=\left|\frac{\mathrm{m}_{1-\mathrm{m}_{2}}}{1+\mathrm{m}_{1} \mathrm{~m}_{2}}\right|$
$\Rightarrow \tan \theta=\left|\frac{\infty-0}{1+\infty \times 0}\right|$
$\Rightarrow \tan \theta=\infty$
$\Rightarrow \theta=\tan ^{-1}(\infty)$
$\therefore \tan ^{-1}(\infty)=\frac{\pi}{2}$
$\Rightarrow \theta=\frac{\pi}{2}$
when $m_{1}=\frac{1}{2} \& m_{2}=2$
$\Rightarrow \tan \theta=\left|\frac{3-1}{1+3 \times 1}\right|$
$\Rightarrow \tan \theta=\left|\frac{2}{4}\right|$
$\Rightarrow \tan \theta=\left|\frac{1}{2}\right|$
$\Rightarrow \theta=\tan ^{-1}\left(\frac{1}{2}\right)$
$\Rightarrow \theta \cong 25.516$
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All Study Material
- JEE Main
- Exam Pattern
- Previous Year Papers
- PYQ Chapterwise
- Physics
- Kinematics 1D
- Kinemetics 2D
- Friction
- Work, Power, Energy
- Centre of Mass and Collision
- Rotational Dynamics
- Gravitation
- Calorimetry
- Elasticity
- Thermal Expansion
- Heat Transfer
- Kinetic Theory of Gases
- Thermodynamics
- Simple Harmonic Motion
- Wave on String
- Sound waves
- Fluid Mechanics
- Electrostatics
- Current Electricity
- Capacitor
- Magnetism and Matter
- Electromagnetic Induction
- Atomic Structure
- Dual Nature of Matter
- Nuclear Physics
- Radioactivity
- Semiconductors
- Communication System
- Error in Measurement & instruments
- Alternating Current
- Electromagnetic Waves
- Wave Optics
- X-Rays
- All Subjects
- Physics
- Motion in a Plane
- Law of Motion
- Work, Energy and Power
- Systems of Particles and Rotational Motion
- Gravitation
- Mechanical Properties of Solids
- Mechanical Properties of Fluids
- Thermal Properties of matter
- Thermodynamics
- Kinetic Theory
- Oscillations
- Waves
- Electric Charge and Fields
- Electrostatic Potential and Capacitance
- Current Electricity
- Thermoelectric Effects of Electric Current
- Heating Effects of Electric Current
- Moving Charges and Magnetism
- Magnetism and Matter
- Electromagnetic Induction
- Alternating Current
- Electromagnetic Wave
- Ray Optics and Optical Instruments
- Wave Optics
- Dual Nature of Radiation and Matter
- Atoms
- Nuclei
- Semiconductor Electronics: Materials, Devices and Simple Circuits.
- Chemical Effects of Electric Current,