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.
ScanSolve AI Answer
Hey, good to see you again. The problem asks to find the equation of the tangent and normal to the curve x^2 + y^2 = 11 at the point (3, 1). Step 1: Find the derivative (dy)/(dx) using implicit differentiation. The equation of the curve is x^2 + y^2 = 11. Differentiate both sides with respect to x: (d)/(dx)(x^2) + (d)/(dx)(y^2) = (d)/(dx)(11) 2x + 2y (dy)/(dx) = 0 2y (dy)/(dx) = -2x (dy)/(dx) = -(2x)/(2y) (dy)/(dx) = -(x)/(y) Step 2: Calculate the slope of the tangent at the given point (3, 1). Substitute x=3 and y=1 into the derivative: m_tangent = (dy)/(dx) |_(3,1) = -(3)/(1) = -3 Step 3: Find the equation of the tangent line. The equation of a line is y - y_1 = m(x - x_1). Using the point (x_1, y_1) = (3, 1) and slope m_tangent = -3: y - 1 = -3(x - 3) y - 1 = -3x + 9 y = -3x + 10 Rearrange into the standard form Ax + By + C = 0: 3x + y - 10 = 0 Step 4: Calculate the slope of the normal line. The normal line is perpendicular to the tangent line. The product of their slopes is -1. m_normal = -(1)/(m_tangent) m_normal = -(1)/(-3) = (1)/(3) Step 5: Find the equation of the normal line. Using the point (x_1, y_1) = (3, 1) and slope m_normal = (1)/(3): y - 1 = (1)/(3)(x - 3) Multiply by 3 to clear the fraction: 3(y - 1) = x - 3 3y - 3 = x - 3 3y = x Rearrange into the standard form Ax + By + C = 0: x - 3y = 0 The equation of the tangent line is 3x + y - 10 = 0. The equation of the normal line is x - 3y = 0. Send me the next one 📸