91Ó°ÊÓ

To begin unraveling the various aspects of coordinate geometry involved in this problem, let's start with understanding the circle equation given, which is in the form of \(x^{2} + y^{2} = 4\). In coordinate geometry, the general equation of a circle centered at the origin is \(x^{2} + y^{2} = r^2\), where \(r\) is the radius of the circle. The given circle equation shows that our circle has a center at the origin \((0,0)\) and a radius \(r = 2\) (since \(\sqrt{4} = 2\)). This means every point on the circle is exactly 2 units away from the origin. It's crucial to deduce this, as it is a stepping stone for further calculations that involve the circle's geometry.

Next, we need to explore the interaction between this circle and the line joining points \(A(1,0)\) and \(B(3,4)\). A line’s equation usually requires both a slope and a point. The formula for slope \(m\) between two points \((x_1, y_1)\) and \((x_2, y_2)\) is \(m = \frac{y_2 - y_1}{x_2 - x_1}\). Here, it computes to \(m = 2\).
The point-slope form is a reliable method to construct the line equation, given by \(y - y_1 = m(x - x_1)\). For point \(A(1,0)\), the equation becomes \(y = 2(x - 1)\), simplifying to \(y = 2x - 2\). This tells us that the line extends infinitely in both directions through these points and might intersect the circle at specific points, called intersections.

To find where the line intersects the circle, substitute \(y = 2x - 2\) from the line equation into the circle equation \(x^2 + y^2 = 4\). The substitution transforms it into a single-variable quadratic equation: \(x^2 + (2x - 2)^2 = 4\). Simplifying further, it leads to \(5x^2 - 8x = 0\).

• This equation, \(5x^2 - 8x = 0\), utilizes the property of quadratic equations, \(ax^2 + bx + c = 0\).
• On factoring, it splits into \(x(5x - 8) = 0\), revealing potential intersection points \(x = 0\) or \(x = \frac{8}{5}\).

This equation allows us to find specific values of \(x\), which then provide corresponding \(y\) values when substituted back into the line equation. Each solution gives an intersection point where the circle and the line meet.

To relate the proportions \(\alpha\) and \(\beta\), distances between various points need to be determined. This is where the distance formula becomes essential. The formula for the distance \(d\) between any two points \((x_1, y_1)\) and \((x_2, y_2)\) is \(d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}\).
Utilizing this:

• The distance \(BP\) for point \(P(0, -2)\) from \(B(3,4)\) is \(\sqrt{(3-0)^2 + (4+2)^2} = \sqrt{45}\).
• Similarly, distance \(PA\) for \(A(1,0)\) from \(P\) is \(\sqrt{(1-0)^2 + (0+2)^2} = \sqrt{5}\).

These distances help compute the ratio \(\alpha = \frac{BP}{PA} = 3\). Performing similar calculations for \(Q\) and analyzing the relationships between these distances allows us to determine \(\beta\), supporting the formation of a quadratic equation based on these ratios.