Problem 167 - 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 (A) \(-\bar{z}_1 z_2 z_3\).

Step 1: Understand Problem Context

We are given three points \( A, B, \) and \( C \) that represent the complex numbers \( z_1, z_2, z_3 \), respectively. These points form a triangle in the complex plane whose circumcentre lies at the origin. The altitude from point \( A \) to side \( BC \), denoted by \( AD \), intersects the circumcircle again at a point \( P \). Our task is to determine the complex number that represents \( P \).

Step 2: Utilize the Circumcentre Property

Since the circumcentre of triangle \( ABC \) lies at the origin, it implies that the circle through \( A, B, \) and \( C \) is centered at the origin. This means \[ |z_1| = |z_2| = |z_3|. \] This is a crucial property because \( A, B, \) and \( C \) are all equidistant from the origin.

Step 3: Identify the Reflection Property

The point \( P \), where the altitude \( AD \) meets the circumcircle again, is the reflection of \( A \) across the line \( BC \). In complex terms, reflection involves conjugation and transformation.

Step 4: Calculate the Complex Number for P

For a point \( Q \) in the complex plane, the reflection over the origin along any line \(through \(Q\)\) is represented by \(-\bar{z}\). Hence, based on symmetry and given that the origin is the circumcentre, point \( P \) is the reflection of \( A \) across the line \( BC \). Therefore, the complex number for \( P \) is \(-\bar{z}_1 z_2 z_3\), which matches option (A).

Step 5: Verify with Given Options

We derived that \( P \) represents the complex number \(-\bar{z}_1 z_2 z_3\). Check this against the given options: (A) \(-\bar{z}_1 z_2 z_3\), (B) \(-\frac{\bar{z}_1 z_2}{\bar{z}_3}\), (C) \(-\frac{\bar{z}_1 z_3}{\bar{z}_2}\), (D) \(-\frac{z_2 z_3}{z_1}\). Clearly, the answer matches option (A).

Key Concepts

Circumcentre

The circumcentre of a triangle is a pivotal concept in geometry and has intriguing implications when applied to the complex plane. In traditional geometry, the circumcentre is the point equidistant from all vertices of a triangle. This property implies that it is the centre of the circumcircle, which passes through all three vertices of the triangle. In the complex plane, this concept is adapted using complex numbers to represent point coordinates. For the triangle formed by points corresponding to complex numbers \(z_1, z_2,\) and \(z_3\), having the circumcentre at the origin is quite significant. This condition means that the distances from the origin to each vertex are equal:

• \(|z_1| = |z_2| = |z_3|\)

This equality forms a geometric circle centered at the origin, through which all the vertices lie. Understanding this can simplify the process of solving problems involving reflection and geometric properties of triangles in the complex plane.

Reflection Property

In geometry, reflection is a transformation creating a mirror image of a point or shape across a specific line or plane. In the realm of complex numbers, it is realized through conjugation and transformation of the components of the complex number. In the problem concerning the triangle \(ABC\), reflection is utilized to find the point \(P\). Given that the line \(AD\) is an altitude, the problem states \(P\) is where this line intersects the circumcircle again. Because the circumcentre is at the origin, the reflection properties simplify based on symmetry. When reflecting a complex number \(z\) across a line, or in our problem, where point \(A\) reflects over line \(BC\), this reflection is typically expressed with complex conjugates. The resulting reflection point, such as \(P\), is represented in complex terms as:

• \(P = -\bar{z}_1 z_2 z_3\)

Identifying such reflections can help in analyzing the geometry of the triangle and understanding symmetric relationships and transformations in the complex plane.

Triangle Geometry

Triangle geometry on the complex plane combines traditional geometric principles with algebraic manipulations involving complex numbers. Such geometry is not merely an abstraction but provides real, applicable insights into geometric problem-solving.In triangle \(ABC\) represented by complex numbers \(z_1, z_2,\) and \(z_3\), several core principles guide our understanding:

• Equilateral or isosceles properties can be determined directly through equal magnitudes like \(|z_1| = |z_2| = |z_3|\) from the circumcentre condition.
• Altitudes, lines that drop from one vertex and are perpendicular to the opposite side, help in finding various important points such as incenters, centroids, and orthocenters.
• The circumcircle itself, passing through all vertices, reinforces the notion that these geometric constructions have concrete, measurable properties in terms of distance (in this case radius from the circumcentre).

By working with complex numbers, one can derive significant results based on these geometric principles, providing powerful tools for solving complex geometric configurations.