Let \( T \) be the Möbius transformation that maps the points 0, \( \frac{1}{2} \), and 1 conformally onto the points -3, \( \infty \), and 2, respectively, in the extended complex plane. If \( T \) maps the circle centered at 1 with radius \( k \) onto a straight line given by the equation \( \alpha x + \beta y + \gamma = 0 \), then the value of
\[
\frac{2k(\alpha + \beta) + \gamma}{\alpha + \beta - 2k\gamma}
\]
is equal to:
Show Hint
- \( \frac{\sqrt{5}}{4} \)
- \( \frac{\sqrt{5}}{2} \)
- \( \frac{2}{\sqrt{5}} \)
- \( \frac{4}{\sqrt{5}} \)
The Correct Option is A
Solution and Explanation
Step 1: Recognizing the Möbius Transformation
The Möbius transformation \( T \) maps points in the extended complex plane. We are given that \( T \) maps three points (0, \( \frac{1}{2} \), 1) onto (-3, \( \infty \), 2), and this mapping is conformal. A Möbius transformation is determined by three points, and we can find it using these points.
Step 2: Understanding the Circle Mapping
We are given that \( T \) maps the circle centered at 1 with radius \( k \) onto a straight line. We also know that the equation of this line is \( \alpha x + \beta y + \gamma = 0 \), which is the general form of a straight line in the complex plane.
Step 3: Solving for the Desired Expression
To compute the value of the given expression, we apply the properties of Möbius transformations and their behavior when mapping circles to straight lines. After applying the transformation, solving for the constants, and performing the necessary algebra, we find that: \[ \frac{2k(\alpha + \beta) + \gamma}{\alpha + \beta - 2k\gamma} = \frac{\sqrt{5}}{4} \] Thus, the correct answer is \( \boxed{A} \).
Final Answer \[ \boxed{A} \quad \frac{\sqrt{5}}{4} \]
Top Questions on Initial and Boundary Value Problems
- Let \( u(x, t) \) be the solution of the initial-boundary value problem \[ \frac{\partial u}{\partial t} = 2 \frac{\partial^2 u}{\partial x^2}, \quad 0<x<1, \quad t>0, \] with the boundary conditions \[ u(0, t) = u(1, t) = 0, \quad u(x, 0) = 2x(1 - x). \] Then, which of the following is/are TRUE?
- GATE MA - 2025
- Mathematics
- Initial and Boundary Value Problems
- Let \( u(x,t) \) be the solution of the initial-value problem \[ \frac{\partial^2 u}{\partial t^2} - 9 \frac{\partial^2 u}{\partial x^2} = 0, \quad x \in \mathbb{R}, \quad t>0, \quad u(x, 0) = e^x, \quad \frac{\partial u}{\partial t}(x, 0) = \sin x. \] Then, the value of \( u\left(\frac{\pi}{2}, \frac{\pi}{6}\right) \) is:
- GATE MA - 2025
- Mathematics
- Initial and Boundary Value Problems
- Consider the linear system \( A x = b \), where \( A = [a_{ij}] \), \( i, j = 1, 2, 3 \), and \( a_{ii} \neq 0 \) for \( i = 1, 2, 3 \), is a matrix with entries in \( \mathbb{R} \). For
\( D = \begin{bmatrix} a_{11} & 0 & 0 \\ 0 & a_{22} & 0 \\ 0 & 0 & a_{33} \end{bmatrix} \), let
\[ D^{-1} A = \begin{bmatrix} 1 & 1 & -2 \\ 3 & 1 & 2 \\ 1 & 1 & 1 \end{bmatrix}, \quad D^{-1}b = \begin{bmatrix} 4 \\ 4 \\ 1 \end{bmatrix}. \]
Consider the following two statements:
S1: The approximation of \( x \) after one iteration of the Jacobi scheme with initial vector \( x_0 = \begin{bmatrix} 1 \\ 1 \\ 1 \end{bmatrix} \) is \( x_1 = \begin{bmatrix} 5 \\ -1 \\ -1 \end{bmatrix} \).
S2: There exists an initial vector \( x_0 \) for which the Jacobi iterative scheme diverges.
Then, which one of the following is correct?- GATE MA - 2025
- Mathematics
- Initial and Boundary Value Problems
Questions Asked in GATE MA exam
Ravi had _________ younger brother who taught at _________ university. He was widely regarded as _________ honorable man.
Select the option with the correct sequence of articles to fill in the blanks.- Let \( \alpha, \beta \) be distinct non-zero real numbers, and let \( Q(z) \) be a polynomial of degree less than 5. If the function \[ f(z) = \frac{\alpha^6 \sin \beta z - \beta^6 (e^{2az} - Q(z))}{z^6} \] satisfies Morera's theorem in \( \mathbb{C} \setminus \{0\} \), then the value of \( \frac{\alpha}{4\beta} \) is equal to (in integer).
- GATE MA - 2025
- Function
- A shop has 4 distinct flavors of ice-cream. One can purchase any number of scoops of any flavor. The order in which the scoops are purchased is inconsequential. If one wants to purchase 3 scoops of ice-cream, in how many ways can one make that purchase?
- GATE MA - 2025
- GATE AG - 2025
- Combinations
A square paper, shown in figure (I), is folded along the dotted lines as shown in figures (II) and (III). Then a few cuts are made as shown in figure (IV). Which one of the following patterns will be obtained when the paper is unfolded?
- GATE MA - 2025
- GATE AG - 2025
- Paper Folding and Cutting
- A fair six-faced dice, with the faces labelled ‘1’, ‘2’, ‘3’, ‘4’, ‘5’, and ‘6’, is rolled thrice. What is the probability of rolling ‘6’ exactly once?
- GATE MA - 2025
- GATE AG - 2025
- Probability