This mathematics problem involves applying core mathematical principles and formulas. Below you will find a complete step-by-step solution with detailed explanations for each step, helping you understand not just the answer but the method behind it.
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Coordinate Geometry 1. Find equation of line through (1,2) and (3,6) Step 1: Calculate the slope (m) of the line using the formula m = (y_2 - y_1)/(x_2 - x_1). Given points are (x_1, y_1) = (1,2) and (x_2, y_2) = (3,6). m = (6 - 2)/(3 - 1) = (4)/(2) = 2 Step 2: Use the point-slope form of a linear equation, y - y_1 = m(x - x_1). Using point (1,2) and slope m=2: y - 2 = 2(x - 1) y - 2 = 2x - 2 y = 2x - 2 + 2 y = 2x The equation of the line is y = 2x. 2. Find distance between (2,3) and (6,7) Step 1: Use the distance formula, D = sqrt((x_2 - x_1)^2 + (y_2 - y_1)^2). Given points are (x_1, y_1) = (2,3) and (x_2, y_2) = (6,7). D = sqrt((6 - 2)^2 + (7 - 3)^2) D = sqrt((4)^2 + (4)^2) D = sqrt(16 + 16) D = sqrt(32) D = sqrt(16 × 2) D = 4sqrt(2) The distance between the points is 4sqrt(2) units. Circle Geometry 1. Find equation of circle with center (0,0), radius 5 Step 1: The standard equation of a circle with center (h,k) and radius r is (x-h)^2 + (y-k)^2 = r^2. Given center (h,k) = (0,0) and radius r=5. Substitute these values into the equation: (x-0)^2 + (y-0)^2 = 5^2 x^2 + y^2 = 25 The equation of the circle is x^2 + y^2 = 25. 2. State equation of tangent to x^2 + y^2 = 25 at (3,4) Step 1: The equation of the circle is x^2 + y^2 = 25. This means the center is (0,0) and r^2 = 25. The point of tangency is (x_1, y_1) = (3,4). Step 2: For a circle centered at the origin, the equation of the tangent at a point (x_1, y_1) on the circle is given by x x_1 + y y_1 = r^2. Substitute x_1=3, y_1=4, and r^2=25: x(3) + y(4) = 25 3x + 4y = 25 The equation of the tangent is 3x + 4y = 25.