91Ó°ÊÓ

To find the distance between two points in a plane, we use the distance formula, which is extremely useful in geometry. The formula is derived from the Pythagorean theorem and is expressed as: \[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \]This formula calculates the straight-line distance between two points, say, \( P(x_1, y_1) \) and \( Q(x_2, y_2) \). It involves finding the difference in the \( x \)-coordinates and the difference in the \( y \)-coordinates, squaring both differences, adding them, and finally taking the square root.
For example, in the context of our problem, where points \( P \) and \( Q \) are located in the first quadrant, we calculate their distance by plugging their coordinates into this formula.

• The x-coordinates are the same, so \( x_2 - x_1 = 0 \).
• We only focus on the y-coordinate difference, \( y - \frac{1}{2}x \).

This simplifies to finding \( |y - \frac{1}{2}x| \), and results in the expression for distance \( PQ \). Remember that the distance should always be non-negative, hence the absolute value operation ensuring this.

Asymptotes are lines that a curve approaches as it moves towards infinity. For hyperbolas, they play a critical role in defining the graph's shape because the hyperbola will curve around its asymptotes but never intersect them.
The asymptotes of a hyperbola in standard form \( \frac{x^2}{a^2} - \frac{y^2}{b^2} = 1 \) are given by the equations \( y = \pm \frac{b}{a}x \).In our context, the hyperbola \( \frac{x^2}{4} - y^2 = 1 \) leads to asymptotes \( y = \pm \frac{1}{2}x \), where \( a = 2 \) and \( b = 1 \). These lines help establish positions on the graph, such as the point \( Q \), which shares its x-coordinate with \( P \) and lies on the asymptote.
In graphical terms:

• Asymptotes cross at the center of the hyperbola, here it's the origin \((0,0)\).
• The hyperbola's branches approach these lines but will never cross.

For point \( Q \), we use the positive slope of the asymptote \( \frac{1}{2}x \), placing it precisely in the first quadrant where both x and y are positive.

The first quadrant of the Cartesian coordinate system is a key area in mathematics, providing a layout where both x and y coordinates are positive. This region is defined by the top right section of the graph, extending into positive x and positive y directions.
Why focus on the first quadrant in our problem? Because both points \( P \) and \( Q \) are restricted to this area. This restriction implies certain properties:

• Both coordinates at any point (x, y) must be positive values.
• This is important for ensuring that the equations and solutions behave as expected, without needing adjustments for negative square roots or coordinates.

In our specific problem, this ensures simplicity in calculations as negative values or reflections into other quadrants are automatically excluded. It simplifies finding distances, as only the positive solutions are valid.