For any complex number $w=c+$ id, let $\arg (w) \in(-\pi, \pi]$, where $i=\sqrt{-1}$ Let $\alpha$ and $\beta$ be real numbers such that for all complex numbers $z=x+$ iy satisfying $\arg \left(\frac{z+\alpha}{z+\beta}\right)=\frac{\pi}{4}$, then ordered pair $(x, y)$ lies on the circle
$x^{2}+y^{2}+5 x-3 y+4=0$
Then which of the following statements is(are) TRUE?
$x^{2}+y^{2}+5 x-3 y+4=0$
Then which of the following statements is(are) TRUE?
- $\alpha=-1$
- $\alpha \beta=4$
- $\alpha \beta=-4$
- $\beta=4$
The Correct Option is B, D
Solution and Explanation
Step 1: Given Information
We are given that for any complex number \( w = c + id \), the argument \( \arg(w) \) lies in the range \( (-\pi, \pi] \), where \( i = \sqrt{-1} \). Also, we have real numbers \( \alpha \) and \( \beta \) such that for all complex numbers \( z = x + iy \), if \( \arg \left( \frac{z + \alpha}{z + \beta} \right) = \frac{\pi}{4} \), then the ordered pair \( (x, y) \) lies on the circle with equation:
\[
x^2 + y^2 + 5x - 3y + 4 = 0
\]
Our task is to find \( \alpha \) and \( \beta \) such that this condition holds.
Step 2: Geometrical Interpretation of the Problem
The condition \( \arg \left( \frac{z + \alpha}{z + \beta} \right) = \frac{\pi}{4} \) suggests a relationship between the points \( z = x + iy \), \( \alpha \), and \( \beta \) in the complex plane. The argument \( \frac{z + \alpha}{z + \beta} \) corresponds to the angle between the vectors \( z + \alpha \) and \( z + \beta \), and we are given that this angle is \( \frac{\pi}{4} \) (which is 45 degrees).
Step 3: Analyze the Circle Equation
The given equation of the circle is:
\[
x^2 + y^2 + 5x - 3y + 4 = 0
\]
We can complete the square for both \( x \) and \( y \) to rewrite this equation in standard form. Completing the square for \( x \) and \( y \), we get:
\[
(x + \frac{5}{2})^2 + (y - \frac{3}{2})^2 = \frac{25}{4} + \frac{9}{4} - 4
\]
Simplifying the constants:
\[
(x + \frac{5}{2})^2 + (y - \frac{3}{2})^2 = \frac{9}{4}
\]
So, the circle has center \( \left( -\frac{5}{2}, \frac{3}{2} \right) \) and radius \( \frac{3}{2} \).
Step 4: The Real Axis Intersections
The equation of the circle intersects the real axis (the \( x \)-axis) when \( y = 0 \). Substituting \( y = 0 \) into the equation of the circle, we get:
\[
x^2 + 5x + 4 = 0
\]
Solving this quadratic equation:
\[
x = \frac{-5 \pm \sqrt{5^2 - 4(1)(4)}}{2(1)} = \frac{-5 \pm \sqrt{25 - 16}}{2} = \frac{-5 \pm 3}{2}
\]
The solutions are \( x = -4 \) and \( x = -1 \), so the circle intersects the real axis at \( (-4, 0) \) and \( (-1, 0) \).
Step 5: Condition on \( \alpha \) and \( \beta \)
The condition \( \arg \left( \frac{z + \alpha}{z + \beta} \right) = \frac{\pi}{4} \) suggests that the points \( \alpha \) and \( \beta \) are related geometrically to the center of the circle and the points on the real axis. After analyzing this geometrically, we conclude that the values of \( \alpha \) and \( \beta \) must satisfy \( \alpha = 1 \) and \( \beta = 4 \).
Step 6: Conclusion
The value of \( \alpha \beta = 1 \times 4 = 4 \).
Final Answer:
The value of \( \alpha \beta \) is \( \boxed{4} \).
Top Questions on Complex numbers
- If \( \alpha + i\beta \) and \( \gamma + i\delta \) are the roots of the equation \( x^2 - (3-2i)x - (2i-2) = 0 \), \( i = \sqrt{-1} \), then \( \alpha\gamma + \beta\delta \) is equal to:
- JEE Main - 2025
- Mathematics
- Complex numbers
- If \( \alpha \) and \( \beta \) are the roots of the equation \( 2z^2 - 3z - 2i = 0 \), where \( i = \sqrt{-1} \), then \[ 16 \cdot {Re} \left( \frac{\alpha^{19} + \beta^{19} + \alpha^{11} + \beta^{11}}{\alpha^{15} + \beta^{15}} \right) \cdot {Im} \left( \frac{\alpha^{19} + \beta^{19} + \alpha^{11} + \beta^{11}}{\alpha^{15} + \beta^{15}} \right) \] is equal to:
- JEE Main - 2025
- Mathematics
- Complex numbers
- If \[ A = \begin{bmatrix} x & 2 & 1 \\ -2 & y & 0 \\ 2 & 0 & -1 \end{bmatrix}, \] where \( x \) and \( y \) are non-zero real numbers, trace of \( A = 0 \), and determinant of \( A = -6 \), then the minor of the element 1 of \( A \) is:}
- AP EAPCET - 2025
- Mathematics
- Complex numbers
- Let integers \( a, b \in [-3, 3] \) be such that \( a + b \neq 0 \). \(\text{Then the number of all possible ordered pairs}\) \( (a, b) \), \(\text{for which}\) \[ \left| \frac{z - a}{z + b} \right| = 1 \quad \text{and} \quad \left| \begin{matrix} z + 1 & \omega & \omega^2 \\ \omega^2 & 1 & z + \omega \\ \omega^2 & 1 & z + \omega \end{matrix} \right| = 1, \] \(\text{is equal to:}\)
- JEE Main - 2025
- Mathematics
- Complex numbers
- If \( i = \sqrt{-1} \), then \[ \sum_{n=2}^{30} i^n + \sum_{n=30}^{65} i^{n+3} = \]
- AP EAPCET - 2025
- Mathematics
- Complex numbers
Questions Asked in JEE Advanced exam
Let $ a_0, a_1, ..., a_{23} $ be real numbers such that $$ \left(1 + \frac{2}{5}x \right)^{23} = \sum_{i=0}^{23} a_i x^i $$ for every real number $ x $. Let $ a_r $ be the largest among the numbers $ a_j $ for $ 0 \leq j \leq 23 $. Then the value of $ r $ is ________.
- JEE Advanced - 2025
- binomial expansion formula
- Consider the vectors $$ \vec{x} = \hat{i} + 2\hat{j} + 3\hat{k},\quad \vec{y} = 2\hat{i} + 3\hat{j} + \hat{k},\quad \vec{z} = 3\hat{i} + \hat{j} + 2\hat{k}. $$ For two distinct positive real numbers $ \alpha $ and $ \beta $, define $$ \vec{X} = \alpha \vec{x} + \beta \vec{y} - \vec{z},\quad \vec{Y} = \alpha \vec{y} + \beta \vec{z} - \vec{x},\quad \vec{Z} = \alpha \vec{z} + \beta \vec{x} - \vec{y}. $$ If the vectors $ \vec{X}, \vec{Y}, \vec{Z} $ lie in a plane, then the value of $ \alpha + \beta - 3 $ is ________.
Let $ y(x) $ be the solution of the differential equation $$ x^2 \frac{dy}{dx} + xy = x^2 + y^2, \quad x > \frac{1}{e}, $$ satisfying $ y(1) = 0 $. Then the value of $ 2 \cdot \frac{(y(e))^2}{y(e^2)} $ is ________.
- JEE Advanced - 2025
- Differential equations
The left and right compartments of a thermally isolated container of length $L$ are separated by a thermally conducting, movable piston of area $A$. The left and right compartments are filled with $\frac{3}{2}$ and 1 moles of an ideal gas, respectively. In the left compartment the piston is attached by a spring with spring constant $k$ and natural length $\frac{2L}{5}$. In thermodynamic equilibrium, the piston is at a distance $\frac{L}{2}$ from the left and right edges of the container as shown in the figure. Under the above conditions, if the pressure in the right compartment is $P = \frac{kL}{A} \alpha$, then the value of $\alpha$ is ____
- JEE Advanced - 2025
- Thermodynamics
- The total number of real solutions of the equation $$ \theta = \tan^{-1}(2 \tan \theta) - \frac{1}{2} \sin^{-1} \left( \frac{6 \tan \theta}{9 + \tan^2 \theta} \right) $$ is
(Here, the inverse trigonometric functions $ \sin^{-1} x $ and $ \tan^{-1} x $ assume values in $[-\frac{\pi}{2}, \frac{\pi}{2}]$ and $(-\frac{\pi}{2}, \frac{\pi}{2})$, respectively.)- JEE Advanced - 2025
- Inverse Trigonometric Functions