A, B represent the complex numbers z1, z2 in the complex plane. If O is the origin not lying on AB, \angle AOB = \alpha, OA = OB and z1, z2 are the roots of the equation z^2 + pz + q = 0, where p, q \in C, then p^2 =

The correct answer is Option 1. Full mathematical solution steps and hints are accessible in the interactive StemCET workspace.

Maths Q8 – AP-EAMCET 2026

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Home Maths Complex & QuadraticQuestion #53

A, B represent the complex numbers

z_{1}, z_{2}

in the complex plane. If

O

is the origin not lying on

A B

∠ A O B = α

O A = O B

and

z_{1}, z_{2}

are the roots of the equation

z^{2} + p z + q = 0

, where

p, q \in C

, then

p^{2} =

Clue 1 of 3: Geometric Interpretation of Complex Numbers

Understand how complex numbers $z_{1}$ and $z_{2}$ relate to points A and B in the complex plane, and how conditions like $O A = O B$ and $∠ A O B = α$ translate to properties of $z_{1}$ and $z_{2}$ .

🧠 Full Solution Path

Step 1: Relate Roots to Coefficients of the Quadratic Equation✦ Active

Given the quadratic equation $z^{2} + p z + q = 0$ , and its roots are $z_{1}$ and $z_{2}$ . According to Vieta's formulas, the sum and product of the roots are:

z_{1} + z_{2} = - p (Equation 1)

z_{1} z_{2} = q (Equation 2)

We need to find $p^{2}$ , which can be expressed as $p^{2} = (- p)^{2} = (z_{1} + z_{2})^{2}$ .

💡 Teacher's Secret Hint

Remember that Vieta's formulas are crucial for connecting the roots of a polynomial to its coefficients. Here, we are working with a monic quadratic equation ( $a = 1$ ). Make sure to correctly handle the signs for the sum of roots.

Step 2: Express Roots Using Geometric Conditions○ Expand

The problem states that $A$ and $B$ represent $z_{1}$ and $z_{2}$ respectively, $O$ is the origin, $O A = O B$ , and $∠ A O B = α$ . The condition $O A = O B$ implies that the moduli of $z_{1}$ and $z_{2}$ are equal. Let $| z_{1} | = | z_{2} | = r$ . The condition $∠ A O B = α$ means the angle between the complex numbers $z_{1}$ and $z_{2}$ is $α$ . Without loss of generality, we can choose a coordinate system such that the bisector of $∠ A O B$ lies along the real axis. This allows us to write the complex numbers as:

z_{1} = r e^{- i α / 2}

z_{2} = r e^{i α / 2}

💡 Teacher's Secret Hint

Representing complex numbers in polar form $r e^{i θ}$ is very helpful for problems involving moduli and arguments. When dealing with angles between complex numbers, it's often convenient to rotate the entire system so that one of the arguments or their bisector aligns with an axis.

Step 3: Evaluate the Product of Roots○ Expand

Substitute the expressions for $z_{1}$ and $z_{2}$ from Step 2 into Equation 2 ( $z_{1} z_{2} = q$ ):

q = (r e^{- i α / 2}) (r e^{i α / 2})

q = r^{2} e^{- i α / 2 + i α / 2}

q = r^{2} e^{0}

q = r^{2} (Equation 3)

💡 Teacher's Secret Hint

Remember the property of exponents for complex numbers: $e^{a} \cdot e^{b} = e^{a + b}$ . Also, $e^{0} = 1$ for any exponent.

Step 4: Evaluate the Sum of Roots○ Expand

Substitute the expressions for $z_{1}$ and $z_{2}$ from Step 2 into Equation 1 ( $z_{1} + z_{2} = - p$ ):

- p = r e^{- i α / 2} + r e^{i α / 2}

- p = r (e^{i α / 2} + e^{- i α / 2})

Using Euler's formula, we know that $e^{i θ} + e^{- i θ} = 2 \cos θ$ . Applying this, we get:

- p = r (2 \cos (α / 2)) (Equation 4)

💡 Teacher's Secret Hint

The identity $e^{i θ} + e^{- i θ} = 2 \cos θ$ is a very useful shortcut derived directly from $e^{i θ} = \cos θ + i \sin θ$ and $e^{- i θ} = \cos θ - i \sin θ$ . Make sure to identify $θ = α / 2$ in this context.

Step 5: Calculate

p^{2}

○ Expand

Square both sides of Equation 4 to find $p^{2}$ :

(- p)^{2} = (2 r \cos (α / 2))^{2}

p^{2} = 4 r^{2} \cos^{2} (α / 2)

Now, substitute the value of $r^{2}$ from Equation 3 ( $r^{2} = q$ ) into this expression:

p^{2} = 4 q \cos^{2} (\frac{α}{2})

This matches option 1.

💡 Teacher's Secret Hint

Always double-check that you've used all the given information. Here, the relationship between $p, q$ and $z_{1}, z_{2}$ was established first, and then the geometric properties were used to simplify $z_{1}$ and $z_{2}$ before substitution.

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Maths Question 8 – AP-EAMCET 2026

Clue 1 of 3: Geometric Interpretation of Complex Numbers

🧠 Full Solution Path