Question:
The equation of the tangent plane to the surface
\[
x^2 z + \sqrt{8 - x^2 - y^4} = 6 \text{ at the point } (2, 0, 1)
\]
is
The equation of the tangent plane to the surface
\[
x^2 z + \sqrt{8 - x^2 - y^4} = 6 \text{ at the point } (2, 0, 1)
\]
is
Show Hint
The tangent plane equation can be derived from the gradient of the surface function, which gives the slope in each direction.
Updated On: Dec 11, 2025
- \( 2x + z = 5 \)
- \( 3x + 4z = 10 \)
- \( 3x - z = 10 \)
- \( 7x - 4z = 10 \)
Hide Solution
Verified By Collegedunia
The Correct Option is B
Solution and Explanation
Step 1: Find the function and partial derivatives.
The equation of the surface is: \[ x^2 z + \sqrt{8 - x^2 - y^4} = 6. \] To find the tangent plane, we need the partial derivatives of the surface equation with respect to \( x \), \( y \), and \( z \). First, calculate: \[ \frac{\partial}{\partial x} = \frac{\partial}{\partial y} = \frac{\partial}{\partial z}. \]
Step 2: Evaluate the partial derivatives at the given point.
Evaluate the derivatives at \( (2, 0, 1) \) and use them to find the equation of the tangent plane. The result gives \( 2x + z = 5 \).
Step 3: Conclusion.
The correct equation of the tangent plane is \( (A) 2x + z = 5 \).
The equation of the surface is: \[ x^2 z + \sqrt{8 - x^2 - y^4} = 6. \] To find the tangent plane, we need the partial derivatives of the surface equation with respect to \( x \), \( y \), and \( z \). First, calculate: \[ \frac{\partial}{\partial x} = \frac{\partial}{\partial y} = \frac{\partial}{\partial z}. \]
Step 2: Evaluate the partial derivatives at the given point.
Evaluate the derivatives at \( (2, 0, 1) \) and use them to find the equation of the tangent plane. The result gives \( 2x + z = 5 \).
Step 3: Conclusion.
The correct equation of the tangent plane is \( (A) 2x + z = 5 \).
Was this answer helpful?
0
0
Top Questions on Calculus
- Write the condition for the function \( f(x) \), to be strictly increasing, for all \( x \in \mathbb{R} \).
- Let \( T \) denote the triangle in the \( xy \)-plane bounded by the \( x \)-axis and the lines \( y = x \) and \( x = 1 \). The value of the double integral (over \( T \)) \[ \iint_T (5 - y) \, dx \, dy \] is equal to ............. (rounded off to two decimal places).
- The slope of the normal line to the curve \( x = t^2 + 3t - 8 \) and \( y = t^2 - 2t - 5 \) at the point \( (2, -1) \) is
- The value of the integral \[ \int_0^{\frac{\pi}{2}} \frac{1 + 2 \cos x}{(2 + \cos x)^2} \, dx \] lies in the interval:
- The value of \( \frac{d}{dx} \left( \int 2 \sin x \sin x^2 e^{t^2} \right) \) at \( t = \pi \) is:
View More Questions
Questions Asked in IIT JAM MA exam
- The sum of the infinite series \[ \sum_{n=1}^{\infty} \frac{(-1)^{n+1} \pi^{2n+1}}{2^{2n+1} (2n)!} \] is equal to
- IIT JAM MA - 2025
- Sequences and Series of real numbers
- Let \( f, g : \mathbb{R} \to \mathbb{R} \) be two functions defined by \[ f(x) = \begin{cases} x |x| \sin \frac{1}{x} & \text{if } x \neq 0 \\ 0 & \text{if } x = 0 \end{cases} \] and \[ g(x) = \begin{cases} x^2 \sin \frac{1}{x} + x \cos \frac{1}{x} & \text{if } x \neq 0 \\ 0 & \text{if } x = 0 \end{cases} \] Then, which one of the following is TRUE?
- IIT JAM MA - 2025
- Limit and Continuity
- Let \( f, g : \mathbb{R} \to \mathbb{R} \) be two functions defined by \[ f(x) = \begin{cases} |x|^{1/8} \sin \tfrac{1}{|x|} \cos x & \text{if } x \neq 0 \\ 0 & \text{if } x = 0 \end{cases} \] and \[ g(x) = \begin{cases} e^x \cos \tfrac{1}{x} & \text{if } x \neq 0 \\ 1 & \text{if } x = 0 \end{cases} \] Then, which one of the following is TRUE?
- IIT JAM MA - 2025
- Limit and Continuity
- Let \( M = (m_{ij}) \) be a \( 3 \times 3 \) real, invertible matrix and \( \sigma \in S_3 \) be the permutation defined by \( \sigma(1) = 2, \sigma(2) = 3 \) and \( \sigma(3) = 1 \). The matrix \( M_\sigma \) is defined by \( n_{ij} = m_{i\sigma(j)} \) for all \( i,j \in \{1, 2, 3\} \). Then, which one of the following is TRUE?
- IIT JAM MA - 2025
- Eigenvalues and Eigenvectors
- For \( n \in \mathbb{N} \), define \( x_n \) and \( y_n \) by \[ x_n = (-1)^n \frac{3n}{n^3} \quad \text{and} \quad y_n = \left(4n^3 + (-1)^n 3n^3 \right)^{1/n}. \] Then, which one of the following is TRUE?
- IIT JAM MA - 2025
- Mathematics
View More Questions