91Ó°ÊÓ

To understand this problem, we need to grasp the distance formula. The distance between two points in a plane can be calculated by: \[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \] This formula is derived from the Pythagorean theorem and includes differences along the x and y coordinates.
For example, if you want to find the distance between points \( (1, 2) \) and \( (4, 6) \), you substitute their coordinates into the formula: \[ d = \sqrt{(4 - 1)^2 + (6 - 2)^2} = \sqrt{3^2 + 4^2} = \sqrt{9 + 16} = \sqrt{25} = 5 \] The result here is 5, a rational number in this case. In our exercise, you need to ensure that any distance between a point and any point on the line is rational.

A point \( P = (a, b) \) is rational if both coordinates \( a \) and \( b \) are rational numbers. But what are rational numbers? Rational numbers are numbers that can be expressed as a fraction \( \frac{p}{q} \) where \( p \) and \( q \) are integers and \( q \) is not zero.
Examples of rational numbers include: \[ -3, 0, 1.5 \left( = \frac{3}{2} \right), \frac{4}{7} \] Irrational numbers, by contrast, cannot be written as a fraction of two integers. Examples are \( \sqrt{2} \) and \( \pi \). In our context, we look for rational coordinates so that computations remain precise and manageable.

In the given problem, the expression inside the square root must be a perfect square. Generally, a quadratic equation has the form \[ ax^2 + bx + c = 0 \]. This is essential for finding solutions where the roots can be derived formulaically.
To ensure that the expression we deal with is a quadratic equation that yields rational results, it should be similar to a perfect square, such as \[ (pr + c)^2 \]. When expanded, it should match the structure of the equation in the problem. Identifying and factoring a quadratic equation important because it often simplifies finding rational solutions for variables.
Example: Solving \[ x^2 - 5x + 6 = 0 \], you can factor it as \[ (x - 2)(x - 3) = 0 \]. Thus, \( x \) values are 2 and 3 which are rational.

Finally, let's understand perfect squares. A number is a perfect square if it's an integer squared. For example, \[ 1, 4, 9, 16, 25 \] are perfect squares (having squares \[ 1^2, 2^2, 3^2, 4^2, 5^2 \]). This concept is important as we check if our quadratic term, \[ 170r^2 - 2ar - 26br + a^2 + b^2 \], forms a perfect square.
For a quadratic expression to be a perfect square, it must fit the form \[ (pr + c)^2 \]. When expanded, it should not leave any ambiguous or non-integer components. In our problem, such perfect square properties simplify the rational requirement.
Identifying if the simplified quadratic in distance form equals a perfect square aids in finding rational solutions.