If the locus of $z \in C$, such that Re $\left(\frac{z-1}{2z + i}\right)$+Re $\left(\frac{\bar{z}-1}{2\bar{z}-i}\right)$ = 2, is a circle of radius r and center (a, b), then $\frac{15ab}{r^2}$ is equal to :
Simplifying the Locus Equation
The given equation is: Re $\left(\frac{z-1}{2z + i}\right)$ + Re $\left(\frac{\bar{z}-1}{2\bar{z}-i}\right)$ = 2
Let $w = \frac{z-1}{2z + i}$. We know that $\overline{w} = \frac{\overline{z-1}}{\overline{2z+i}} = \frac{\bar{z}-1}{2\bar{z}-i}$.
The equation simplifies using the property Re($w$) = Re($\overline{w}$): Re($w$) + Re($w$) = 2 $2 \times$ Re($w$) = 2 Re($w$) = 1
Therefore, the condition reduces to finding the locus where Re $\left(\frac{z-1}{2z + i}\right)$ = 1.
Finding the Circle Equation
Let $z = x + iy$. Substitute into the expression:
$z-1 = (x-1) + iy$
$2z+i = 2(x+iy) + i = 2x + i(2y+1)$
Calculate the fraction $\frac{z-1}{2z+i}$:
$\frac{(x-1) + iy}{2x + i(2y+1)} = \frac{((x-1) + iy)(2x - i(2y+1))}{(2x + i(2y+1))(2x - i(2y+1))}$
Numerator = $(x-1)(2x) + y(2y+1) + i(y(2x) - (x-1)(2y+1))$
Numerator = $(2x^2 - 2x + 2y^2 + y) + i(2xy - (2xy + x - 2y - 1))$
Numerator = $(2x^2 - 2x + 2y^2 + y) + i(-x + 2y + 1)$
Denominator = $(2x)^2 + (2y+1)^2 = 4x^2 + 4y^2 + 4y + 1$
The real part is Re $\left(\frac{z-1}{2z+i}\right) = \frac{2x^2 - 2x + 2y^2 + y}{4x^2 + 4y^2 + 4y + 1}$.
Set the real part equal to 1:
$\frac{2x^2 - 2x + 2y^2 + y}{4x^2 + 4y^2 + 4y + 1} = 1$
$2x^2 - 2x + 2y^2 + y = 4x^2 + 4y^2 + 4y + 1$
Rearrange terms:
$2x^2 + 2y^2 + 2x + 3y + 1 = 0$
Divide by 2 to get the standard form:
$x^2 + y^2 + x + \frac{3}{2}y + \frac{1}{2} = 0$
Identifying Circle Parameters (Center and Radius)
Compare $x^2 + y^2 + x + \frac{3}{2}y + \frac{1}{2} = 0$ with the general circle equation $x^2 + y^2 + 2gx + 2fy + c = 0$.
- Center $(a, b) = (-g, -f)$.
- From the equation, $2g = 1 \implies g = \frac{1}{2}$ and $2f = \frac{3}{2} \implies f = \frac{3}{4}$.
- Thus, the center is $(a, b) = (-\frac{1}{2}, -\frac{3}{4})$.
- The radius squared $r^2 = g^2 + f^2 - c$.
- Here, $c = \frac{1}{2}$.
- $r^2 = (\frac{1}{2})^2 + (\frac{3}{4})^2 - \frac{1}{2} = \frac{1}{4} + \frac{9}{16} - \frac{8}{16} = \frac{4+9-8}{16} = \frac{5}{16}$.
Calculating the Final Expression
We need to find the value of $\frac{15ab}{r^2}$.
- $a = -\frac{1}{2}$
- $b = -\frac{3}{4}$
- $r^2 = \frac{5}{16}$
- $ab = (-\frac{1}{2}) \times (-\frac{3}{4}) = \frac{3}{8}$
Substitute these values into the expression:
$\frac{15ab}{r^2} = \frac{15 \times (\frac{3}{8})}{\frac{5}{16}} = \frac{\frac{45}{8}}{\frac{5}{16}}$
$\frac{45}{8} \times \frac{16}{5} = \frac{45}{5} \times \frac{16}{8} = 9 \times 2 = 18$.
Similar Questions
- Let A be a $3 \times 3$ matrix such that $A + A^T = O$. If $A\begin{bmatrix} 1 \\ -1 \\ 0 \end{bmatrix} = \begin{bmatrix} 3 \\ 3 \\ 2 \end{bmatrix}$, $A^2\begin{bmatrix} 1 \\ -1 \\ 0 \end{bmatrix} = \begin{bmatrix} -3 \\ 19 \\ -24 \end{bmatrix}$ and $\det(adj(2 \ adj(A + I))) = (2)^\alpha \cdot (3)^\beta \cdot (11)^\gamma$, $\alpha, \beta, \gamma$ are non-negative integers, then $\alpha + \beta + \gamma$ is equal to _________
- Let $\alpha = \frac{-1 + i\sqrt{3}}{2}$ and $\beta = \frac{-1 - i\sqrt{3}}{2}$, $i = \sqrt{-1}$. If $(7 - 7\alpha + 9\beta)^{20} + (9 + 7\alpha - 7\beta)^{20} + (-7 + 9\alpha + 7\beta)^{20} + (14 + 7\alpha + 7\beta)^{20} = m^{10}$, then $m$ is _________
- Let ABC be a triangle. Consider four points $p_1, p_2, p_3, p_4$ on the side AB, five points $p_5, p_6, p_7, p_8, p_9$ on the side BC, and four points $p_{10}, p_{11}, p_{12}, p_{13}$ on the side AC. None of these points is a vertex of the triangle ABC. Then the total number of pentagons, that can be formed by taking all the vertices from the points $p_1, p_2, ..., p_{13}$, is _________
- If $X = \begin{bmatrix} x \\ y \\ z \end{bmatrix}$ is a solution of the system of equations $AX = B$, where $\text{adj } A = \begin{bmatrix} 4 & 2 & 2 \\ -5 & 0 & 5 \\ 1 & -2 & 3 \end{bmatrix}$ and $B = \begin{bmatrix} 4 \\ 0 \\ 2 \end{bmatrix}$, then $|x + y + z|$ is equal to :
- Let $C_r$ denote the coefficient of $x^r$ in the binomial expansion of $(1 + x)^n$, $n \in \mathbb{N}, 0 \leq r \leq n$. If $P_n = C_0 - C_1 + \frac{2^2}{3} C_2 - \frac{2^3}{4} C_3 + \dots + \frac{(-2)^n}{n+1} C_n$, then the value of $\sum_{n=1}^{25} \frac{1}{P_{2n}}$ equals.
- The number of elements in the relation $R = \{(x, y) : 4x^2 + y^2 < 52, x, y \in \mathbb{Z}\}$ is
- Let $S = \{z \in \mathbb{C} : 4z^2 + \bar{z} = 0\}$. Then $\sum_{z \in S} |z|^2$ is equal to :
Let f and g be functions satisfying $f(x+y) = f(x)f(y), f(1) = 7$ and $g(x+y) = g(xy), g(1) = 1$, for all $x, y \in \mathbb{N}$. If $\sum_{x=1}^{n} \left(\frac{f(x)}{g(x)}\right) = 19607$, then n is equal to :
Let S be the set of the first 11 natural numbers. Then the number of elements in $A = \{B \subseteq S : n(B) \geq 2$ and the product of all elements of B is even is ________.
- Let $S = \frac{1}{25!} + \frac{1}{3!23!} + \frac{1}{5!21!} + \dots$ up to 13 terms. If $13S = \frac{2^k}{n!}, k \in \mathbb{N}$, then $n + k$ is equal to
Important Questions from Algebra
- Let A be a $3 \times 3$ matrix such that $A + A^T = O$. If $A\begin{bmatrix} 1 \\ -1 \\ 0 \end{bmatrix} = \begin{bmatrix} 3 \\ 3 \\ 2 \end{bmatrix}$, $A^2\begin{bmatrix} 1 \\ -1 \\ 0 \end{bmatrix} = \begin{bmatrix} -3 \\ 19 \\ -24 \end{bmatrix}$ and $\det(adj(2 \ adj(A + I))) = (2)^\alpha \cdot (3)^\beta \cdot (11)^\gamma$, $\alpha, \beta, \gamma$ are non-negative integers, then $\alpha + \beta + \gamma$ is equal to _________
- Let $\alpha = \frac{-1 + i\sqrt{3}}{2}$ and $\beta = \frac{-1 - i\sqrt{3}}{2}$, $i = \sqrt{-1}$. If $(7 - 7\alpha + 9\beta)^{20} + (9 + 7\alpha - 7\beta)^{20} + (-7 + 9\alpha + 7\beta)^{20} + (14 + 7\alpha + 7\beta)^{20} = m^{10}$, then $m$ is _________
- Let ABC be a triangle. Consider four points $p_1, p_2, p_3, p_4$ on the side AB, five points $p_5, p_6, p_7, p_8, p_9$ on the side BC, and four points $p_{10}, p_{11}, p_{12}, p_{13}$ on the side AC. None of these points is a vertex of the triangle ABC. Then the total number of pentagons, that can be formed by taking all the vertices from the points $p_1, p_2, ..., p_{13}$, is _________
- If $X = \begin{bmatrix} x \\ y \\ z \end{bmatrix}$ is a solution of the system of equations $AX = B$, where $\text{adj } A = \begin{bmatrix} 4 & 2 & 2 \\ -5 & 0 & 5 \\ 1 & -2 & 3 \end{bmatrix}$ and $B = \begin{bmatrix} 4 \\ 0 \\ 2 \end{bmatrix}$, then $|x + y + z|$ is equal to :
- Let $C_r$ denote the coefficient of $x^r$ in the binomial expansion of $(1 + x)^n$, $n \in \mathbb{N}, 0 \leq r \leq n$. If $P_n = C_0 - C_1 + \frac{2^2}{3} C_2 - \frac{2^3}{4} C_3 + \dots + \frac{(-2)^n}{n+1} C_n$, then the value of $\sum_{n=1}^{25} \frac{1}{P_{2n}}$ equals.
