Find a point on the curve $x y+4=0$ at which the tangents are inclined at an angle of $45^{\circ}$ with the $x$-axis.
Given:
The curve is $x y+4=0$
If a tangent line to the curve $y=f(x)$ makes an angle $\boldsymbol{\theta}$ with $x$-axis in the positive direction, then
$\frac{\mathrm{d} y}{\mathrm{dx}}=$ The Slope of the tangent $=\tan \theta$
$x y+4=0$
Differentiating the above w.r.t $x$
$\Rightarrow x \times \frac{d}{d x}(y)+y \times \frac{d}{d x}(x)+\frac{d}{d x}(4)=0$
$\Rightarrow x \frac{d y}{d x}+y=0$
$\Rightarrow x \frac{d y}{d x}=-y$
$\Rightarrow \frac{d y}{d x}=\frac{-y}{x} \ldots(1)$
Also, $\frac{\mathrm{dy}}{\mathrm{dx}}=\tan 45^{\circ}=1 \ldots(2)$
From (1) \& (2), we get,
$\Rightarrow \frac{-y}{x}=1$
$\Rightarrow x=-y$
Substitute in $x y+4=0$, we get
$\Rightarrow x(-x)+4=0$
$\Rightarrow-x^{2}+4=0$
$\Rightarrow x^{2}=4$
$\Rightarrow x=\pm 2$
so when $x=2, y=-2$
$\&$ when $x=-2, y=2$
Thus, the points are $(2,-2) \&(-2,2)$
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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,