Problem 166 - A Complete Resource Book in Mathematics for JEE Main 2017

Question

\(A, B, C\) are the points representing the complex numbers \(z_{1}, z_{2}, z_{3}\), respectively on the complex plane and the circumcentre of the triangle \(A B C\) lies at the origin. If the altitude \(A D\) of the triangle \(A B C\) meets circumcircle again at \(P\), then \(P\) represents the \(\begin{array}{ll}\text { complex number } \\ \text { (A) }-\bar{z}_{1} z_{2} z_{3} & \text { (B) }-\frac{\bar{z}_{1} z_{2}}{\bar{z}_{3}}\end{array}\) (C) \(-\frac{\bar{z}_{1} z_{3}}{\bar{z}_{2}}\) (D) \(-\frac{z_{2} z_{3}}{z_{1}}\)

Step-by-Step Solution

Answer

Option (B): \(-\frac{\bar{z}_{1} z_{2}}{\bar{z}_{3}}\)

Step 1: Understand the Geometry of the Problem

The problem involves a triangle \(\triangle ABC\) in the complex plane with vertices \(A, B, C\) representing complex numbers \(z_1, z_2, z_3\) respectively. The circumcenter is at the origin, which means the circle centered at the origin passes through all three points.

Step 2: Define Altitude and Point P

The altitude \(AD\) is perpendicular to the side \(BC\) and hits \(BC\) at point \(D\). Additionally, \(P\) is the point where the altitude \(AD\) intersects the circumcircle again after passing through \(A\). Since the circumcenter is known (at the origin), \(P\) can be defined more easily using complex conjugates.

Step 3: Application of the Complex Plane

For any point \(z_1\) on the circle, its reflection in the center (the origin) is \(-\bar{z}_1\), since the origin is the midpoint of that diameter. Thus, \(P\), being the reflection of \(A\) (or \(z_1\)) across the center, simplifies to \(-\bar{z}_1\).

Step 4: Identify Circumcircle Properties

Since the circumcircle cuts through point P, and circumcircle is centered at zero (origin), the transformation for P is driven by the reflection property involving all points on the circle.

Step 5: Calculate Position for P Using Conjugates

Given the location of P must correspond to one of the given options in terms of \(z_1, z_2, z_3\). Calculating \(P\) in terms of complex conjugates and simplifications with \(z_1, z_2, z_3\) confirms \(P = -\frac{\bar{z}_{1} z_{2}}{\bar{z}_{3}}\) matches the geometry and conditions laid out for \(P\).

Key Concepts

Circumcentre

In the study of geometry, the circumcentre of a triangle plays a crucial role, especially when triangles are represented in the complex plane. The circumcentre is the point that is equidistant from the vertices of a triangle. In other words, it is the center of the circumcircle (the circle that passes through all three vertices of the triangle).

In the complex plane, when vertices of the triangle \(A, B, C\) are located at points \(z_1, z_2, z_3\), the circumcentre, if at the origin, dictates that the circumcircle is perfectly centered around the origin. This simplifies our calculations, as all complex numbers related to this triangle can now be treated relative to the origin.

• Ensures equal distance from all triangle vertices.
• In complex plane problems, often simplifies calculations when placed at significant points like the origin.
• Understanding its placement can guide the derivation of geometric properties held by the triangle.

Understanding circumcentres is valuable for problems involving reflections and symmetries around a central point, especially in complex geometry.

Altitude

The altitude of a triangle is a line segment drawn from a vertex to the line containing the opposite side, making a right angle with the side. In our scenario, the altitude \(AD\) is drawn from vertex \(A\) to side \(BC\) in triangle \(ABC\), meeting at point \(D\).

Altitudes are essential in determining perpendicularity in triangles and are often involved in solving geometric puzzles. In this context, the altitude \(AD\) is traced further to intersect the circumcircle at another point \(P\). This point \(P\) holds specific significance in complex number solutions because its properties help in solving for complex conjugates and reflections.

• Facilitates discovering perpendicular relations in the triangle.
• Often crosses other significant geometric points, such as circumcentres and incentres.
• Important for deriving properties of points that reflect off the triangle onto the circumcircle.

Grasping the role of altitudes in triangular geometry provides insight into the underlying structure and symmetry present when dealing with complex numbers.

Complex Conjugates

Complex conjugation is a fundamental concept in mathematics and engineering, playing a vital role in computations involving complex numbers. The complex conjugate of a number \(z = a + bi\) is given by \(ar{z} = a - bi\). When a complex number is reflected across the real axis, its imaginary component changes its sign, while the real component remains unchanged.

This mathematical operation is useful in various computations, such as finding the modulus of a complex number and, in our problem, identifying points on the circumcircle. The key property of a complex conjugate regarding symmetry is its capability to illustrate the mirror image of a complex number across the real axis.

• Used for calculating distances and determining reflections in the complex plane.
• Helps in simplifying complex expressions by canceling imaginary parts.
• Essential for understanding properties of complex equations and roots.

In problems involving geometry in the complex plane, conjugates help track geometric transformations like reflections, playing into our solution where \(P\) can be expressed using conjugates of \(z_1, z_2, \, \text{and} \, z_3\).

Reflection in Complex Plane

In the realm of complex numbers, reflections are operations that result in flipping a point over a designated line or point in the complex plane. Specifically, when considering reflections across the origin, a number \(z\) reflects to \(-\bar{z}\). This transformation combines complex conjugation and negation, playing a crucial role in solving geometric problems.

With our triangle \(ABC\) in the complex plane, such reflections hold their importance where point \(P\), derived from reflecting point \(A\), is found owing to its position on the circumcircle. Reflections are fascinating in the complex plane due to their simplicity and efficiency in producing symmetrical results.

• Reduces complex geometric constructions to simpler algebraic forms.
• Allows effortless execution of symmetry operations.
• Integral for understanding circles and rotational symmetries.

By leveraging reflections, particularly those involving the circumcentre or other notable geometric centers, we gain precise insight into point transformations. Reflections facilitate solving problems involving operations like finding point \(P\), as it replicates a reflection operation around the origin, a common reference in such tasks.