We are given the three vertices of the triangle as the complex numbers $-1 + 7i$, $-1 + xi$, and $3 + 3i$. The right angle is at $-1 + xi$, and the triangle is isosceles, meaning the two other sides are equal in length. To find $x$, we use the distance formula for complex numbers.
1. Calculate the distance between $-1 + 7i$ and $-1 + xi$: The distance between two complex numbers $z_1 = a + bi$ and $z_2 = c + di$ is given by: \[ |z_1 - z_2| = \sqrt{(a - c)^2 + (b - d)^2} \] Applying this to $-1 + 7i$ and $-1 + xi$, the distance is: \[ |(-1 + 7i) - (-1 + xi)| = |7i - xi| = |(7 - x)i| = |7 - x| \] 2. Calculate the distance between $-1 + xi$ and $3 + 3i$: The distance is: \[ |(-1 + xi) - (3 + 3i)| = |-1 - 3 + (x - 3)i| = |-4 + (x - 3)i| \] The magnitude is: \[ \sqrt{(-4)^2 + (x - 3)^2} = \sqrt{16 + (x - 3)^2} \] 3. Since the triangle is isosceles, the two distances must be equal: Set the two distances equal to each other: \[ |7 - x| = \sqrt{16 + (x - 3)^2} \] 4. Square both sides to eliminate the square root: \[ (7 - x)^2 = 16 + (x - 3)^2 \] 5. Expand both sides: \[ (49 - 14x + x^2) = 16 + (x^2 - 6x + 9) \] Simplifying: \[ 49 - 14x + x^2 = 25 + x^2 - 6x \] 6. Cancel $x^2$ from both sides and simplify: \[ 49 - 14x = 25 - 6x \] \[ -14x + 6x = 25 - 49 \] \[ -8x = -24 \] \[ x = 3 \]
The correct option is (B) : \(3\)
Step 1:
Given the vertices are:
\[ A = -1+7i,\quad B = -1+xi,\quad C = 3+3i \] The triangle is right-angled at vertex B, meaning vectors BA and BC are perpendicular.
Step 2 (Vectors BA and BC):
Calculate vectors:
\[ BA = A - B = (-1+7i)-(-1+xi) = (0+(7 - x)i) \]
\[ BC = C - B = (3+3i)-(-1+xi) = (4+(3 - x)i) \]
Step 3 (Condition for perpendicularity):
If vectors are perpendicular, their dot product is zero:
\[ BA \cdot BC = 0 \]
So we have:
\[ (0)(4) + (7 - x)(3 - x) = 0 \]
Simplifying, we get:
\[ (7 - x)(3 - x) = 0 \]
This gives solutions:
\[ x = 7,\quad x = 3 \]
Step 4 (Condition for isosceles triangle):
Since it's an isosceles right-angled triangle, sides BA and BC must have equal lengths:
\[ |BA| = |BC| \]
Compute lengths:
\[ |BA| = |7 - x| \]
\[ |BC| = \sqrt{4^2+(3 - x)^2} = \sqrt{16+(3 - x)^2} \]
Equating lengths:
\[ |7 - x| = \sqrt{16+(3 - x)^2} \]
Square both sides:
\[ (7 - x)^2 = 16+(3 - x)^2 \]
Expand and simplify:
\[ 49 - 14x + x^2 = 16 + 9 - 6x + x^2 \]
Further simplification gives:
\[ 49 - 14x = 25 - 6x \]
Solve for \( x \):
\[ 49 - 25 = 14x - 6x \]
\[ 24 = 8x \]
\[ x = 3 \]
Therefore, the correct answer is:
𝑥 = 3 x=3
Let \( z \) satisfy \( |z| = 1, \ z = 1 - \overline{z} \text{ and } \operatorname{Im}(z)>0 \)
Then consider:
Statement-I: \( z \) is a real number
Statement-II: Principal argument of \( z \) is \( \dfrac{\pi}{3} \)
Then:
If \( z \) and \( \omega \) are two non-zero complex numbers such that \( |z\omega| = 1 \) and
\[ \arg(z) - \arg(\omega) = \frac{\pi}{2}, \]
Then the value of \( \overline{z\omega} \) is: