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
To find the equation of the tangent to the circle x^2 + y^2 + 2x + 4y - 12 = 0 at the point (3, -1), we will first find the center of the circle and then use the fact that the tangent is perpendicular to the radius at the point of tangency. Step 1: Find the center of the circle. The general equation of a circle is x^2 + y^2 + 2gx + 2fy + c = 0, where the center is (-g, -f). Comparing the given equation x^2 + y^2 + 2x + 4y - 12 = 0 with the general form: 2g = 2 g = 1 2f = 4 f = 2 The center of the circle is C(-g, -f) = C(-1, -2). Step 2: Find the slope of the radius. The radius connects the center C(-1, -2) to the point of tangency P(3, -1). The slope of the radius m_r is given by: m_r = (y_2 - y_1)/(x_2 - x_1) = (-1 - (-2))/(3 - (-1)) = (-1 + 2)/(3 + 1) = (1)/(4) Step 3: Find the slope of the tangent line. The tangent line is perpendicular to the radius at the point of tangency. If m_t is the slope of the tangent, then m_t · m_r = -1. m_t · (1)/(4) = -1 m_t = -4 Step 4: Find the equation of the tangent line. Using the point-slope form of a linear equation, y - y_1 = m(x - x_1), with the point P(3, -1) and slope m_t = -4: y - (-1) = -4(x - 3) y + 1 = -4x + 12 y = -4x + 12 - 1 y = -4x + 11 Rearranging into the standard form Ax + By + C = 0: 4x + y - 11 = 0 The equation of the tangent to the circle is 4x + y - 11 = 0. Send me the next one 📸