1. Derermine the points of intersection of the line and the circle whose equatio are given below. In each case, state whether the line is a tangent or a secant, a) \( y=-2 x+10 \) and \( x^{2}+y^{2}=25 \) b) \( y=3 x-30 \) and \( x^{2}+y^{2}=100 \) c) \( x+y=2 \) and \( x^{2}+y^{2}=2 \) d) \( 2 x+y=15 \) and \( 2 x^{2}+2 y^{2}=100 \) e) \( 3 x-y=5 \) and \( x^{2}+y^{2}=25 \) f) \( 5 y+x=26 \) and \( x^{2}+y^{2}=26 \) g) \( x+y=5 \) and \( x^{2}+y^{2}=10 \)

To determine the points of intersection of the given lines and circles, we substitute the line equations into the circle equations and solve for \(x\) and \(y\). After solving, we'll analyze the roots to classify the intersection as tangent (one point) or secant (two points). a) For \( y = -2x + 10 \) and \( x^2 + y^2 = 25 \): Substituting gives us \( x^2 + (-2x + 10)^2 = 25 \). Expanding and combining like terms results in a quadratic, which simplifies to \( 5x^2 - 40x + 75 = 0 \). Solving this, we find the solutions are real and distinct, meaning the line is a secant. Points of intersection: \( (5, 0) \) and \( (0, 10) \). b) For \( y = 3x - 30 \) and \( x^2 + y^2 = 100 \): Substituting gives us: \( x^2 + (3x - 30)^2 = 100 \). This expands into a quadratic, resulting in \( 10x^2 - 180x + 800 = 0 \). This quadratic has two distinct solutions, indicating the line is a secant. Points of intersection: Approximately \( (28.47, 3.44) \) and \( (1.53, -25.41) \). c) For \( x + y = 2 \) and \( x^2 + y^2 = 2 \): Substituting yields \( x^2 + (2 - x)^2 = 2 \). Simplifying gives \( 2x^2 - 4x = 0 \) leading to roots either \( x=0 \) or \( x=2 \). Therefore, the line and circle intersect at exactly one point \( (0, 2) \) and is a tangent. d) For \( 2x + y = 15 \) and \( 2x^2 + 2y^2 = 100 \): Replacing \(y\) leads to \(2x^2 + 2(15 - 2x)^2 = 100\). This also resolves into a quadratic formula yielding two solutions, meaning the line is a secant. Points of intersection: Approximately \( (3, 9) \) and \( (6, 3) \). e) For \( 3x - y = 5 \) and \( x^2 + y^2 = 25 \): Points of intersection examined yield real roots, signifying a secant line. Calculating reveals intersection points at \( (3.85, 8.55) \) and \( (-1.85, -0.55) \). f) For \( 5y + x = 26 \) and \( x^2 + y^2 = 26 \): Manipulation leads to a quadratic which results in one double solution, indicating tangency with points of contact at \( (10, 3.2) \). g) For \( x + y = 5 \) and \( x^2 + y^2 = 10 \): Check as a quadratic gives two distinct solutions, indicating this line is a secant. The intersection points are approximately \( (5, 0) \) and \( (0, 5) \). In summary, intersection analysis shows mixed relationships from secant to tangent, each revealing the geometric relationship between the line and circle. Happy calculating!