Find the co-ordinates of the point on the curve √x + √y = 4 at which tangent is equally inclined to the axes.
Equation of the curve is given by, √x + √y = 4
Now, let (x1, y1) be he required point on the curve
So, √x1 + √y1 = 4
On differentiating on both the sides w.r.t. x1, we get
$\frac{d}{d x_{1}} \sqrt{x_{1}}+\frac{d}{d x_{1}} \sqrt{y_{1}}=\frac{d}{d x_{1}}(4)$
$\frac{1}{2 \sqrt{x_{1}}}+\frac{1}{2 \sqrt{y_{1}}} \cdot \frac{d y_{1}}{d x_{1}}=0$
$\Rightarrow \frac{1}{\sqrt{x_{1}}}+\frac{1}{\sqrt{y_{1}}} \cdot \frac{d y_{1}}{d x_{1}}=0 \Rightarrow \frac{d y_{1}}{d x_{1}}=-\frac{\sqrt{y_{1}}}{\sqrt{x_{1}}}$........$.(i)$
Since the tangent to the given curve at $\left(x_{1}, y_{1}\right)$ is equally inclined to the axes.
$\therefore$ Slope of the tangent $\frac{d y_{1}}{d x}=\pm \tan \frac{\pi}{4}=\pm 1$
So, from eq. (i) we get
$-\frac{\sqrt{y_{1}}}{\sqrt{x_{1}}}=\pm 1$
So, from eq. $(i)$ we get
$-\frac{\sqrt{y_{1}}}{\sqrt{x_{1}}}=\pm 1$
On squaring on both the sides, we get
y1/x1 = 1 ⇒ y1 = x1
Now, putting the value of y1 in the given equation of the curve.
√x1 + √y1 = 4
√x1 + √x1 = 4 ⇒ 2√x1 = 4 ⇒ √x1 = 2 ⇒ x1 = 4
As y1 = x1
⇒ y1 = 4
Therefore, the required point is (4, 4).
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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,