Reciprocal Roots of a Quadratic Equation

To understand the given problem, let's first review some basic concepts related to quadratic equations.

A quadratic equation is a polynomial equation of degree 2, which can be written in the form:
px^2 + qx + r = 0

The roots of a quadratic equation are the values of x for which the equation becomes true. If the roots of a quadratic equation are α and β, then the equation can be factored as:
(x - α)(x - β) = 0

In this problem, we are given that the roots of the equation px^2 + x + r = 0 are reciprocal to each other. Let's assume the roots to be α and 1/α.

Reciprocal Roots

The concept of reciprocal roots states that if α is a root of a quadratic equation, then 1/α is also a root of the equation. Similarly, if 1/α is a root of the equation, then α is also a root of the equation.

Using this concept, we can write the equation with the given reciprocal roots as:
(x - α)(x - 1/α) = 0

Expanding this equation, we get:
x^2 - (α + 1/α)x + 1 = 0

Comparing this equation with the original equation px^2 + x + r = 0, we can conclude the following:

1. The coefficient of x in both equations is 1. Therefore, α + 1/α = 1.
2. The constant term in both equations is r. Therefore, 1 = r.

Solving the Equation

Now, let's use the information obtained from the comparison to solve the equation.

1. α + 1/α = 1
Multiplying both sides by α, we get:
α^2 + 1 = α

Rearranging the terms, we get:
α^2 - α + 1 = 0

2. 1 = r

Now, we can use the given options to find the correct answer.

Analysis of Options

a) p = 2r
This option does not match the derived equations.

b) p = r
This option matches the derived equations, as the constant term r is equal to 1.

c) 2p = r
This option does not match the derived equations.

d) p = 4r
This option does not match the derived equations.

Therefore, the correct answer is option 'B': p = r.