Find the condition that the curves 2x = y2 and 2xy = k intersect orthogonally.
It’s seen that the given curves are equation of two circles.
2x = y2 ….. (1) and
2xy = k ….. (2)
We know that, two circles intersect orthogonally if the angle between the tangents drawn to the two circles at the point of their intersection is 90o.
Now, differentiating equations (1) and (2) w.r.t. t, we get
$2.1=2 y \cdot \frac{d y}{d x} \Rightarrow \frac{d y}{d x}=\frac{1}{y} \Rightarrow m_{1}=\frac{1}{y}$
$\left(m_{1}=\right.$ slope of the tangent $)$
$\Rightarrow \quad 2 x y=k$
$\Rightarrow 2\left[x \cdot \frac{d y}{d x}+y \cdot 1\right]=0$
$\therefore$ $\frac{d y}{d x}=-\frac{y}{x} \Rightarrow m_{2}=-\frac{y}{x}$
$\left[m_{2}=\right.$ slope of the other tangent $]$
If the two tangents are perpendicular to each other,
then $\quad m_{1} \times m_{2}=-1$
$\Rightarrow \quad \frac{1}{y} \times\left(-\frac{y}{x}\right)=-1 \Rightarrow \frac{1}{x}=1 \quad \Rightarrow \quad x=1$
$2.1=2 y \cdot \frac{d y}{d x} \Rightarrow \frac{d y}{d x}=\frac{1}{y} \Rightarrow m_{1}=\frac{1}{y}$
$\left(m_{1}=\right.$ slope of the tangent)
$\Rightarrow$ $2 x y=k$
$\Rightarrow 2\left[x \cdot \frac{d y}{d x}+y \cdot 1\right]=0$
$\therefore \quad \frac{d y}{d x}=-\frac{y}{x} \Rightarrow m_{2}=-\frac{y}{x}$
$\left[m_{2}=\right.$ slope of the other tangent]
If the two tangents are perpendicular to each other,
then $\quad m_{1} \times m_{2}=-1$
$\Rightarrow \quad \frac{1}{y} \times\left(-\frac{y}{x}\right)=-1 \Rightarrow \frac{1}{x}=1 \Rightarrow x=1$
Now, solving equations (1) and (2), we have
y = k/2x [From (2)]
Putting the value of y in equation (1),
2x = (k/2x)2 ⇒ 2x = k2/4x2
8x3 = k2
8(1)3 = k2
k2 = 8
Therefore, the required condition is k2 = 8.
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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,