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
Step 1: Determine the coordinates of the centre O and the point of tangency A. From the diagram 3.5, the centre of the circle is O(8, -5). The point where the tangent touches the circle is A(8, -13). Step 2: Calculate the gradient of the radius OA. The radius OA connects the centre O(8, -5) and the point of tangency A(8, -13). The gradient m_OA is given by the formula m = (y_2 - y_1)/(x_2 - x_1). m_OA = (-13 - (-5))/(8 - 8) = (-13 + 5)/(0) = (-8)/(0) Since the denominator is 0, the gradient is undefined. This means the radius OA is a vertical line. Step 3: Determine the equation of the tangent PQ. The tangent line PQ is perpendicular to the radius OA at the point of tangency A. If the radius OA is a vertical line, then the tangent PQ must be a horizontal line. A horizontal line passing through a point (x_1, y_1) has the equation y = y_1. The tangent PQ passes through point A(8, -13). Therefore, the equation of the tangent PQ is y = -13. The equation of the tangent PQ is y = -13. Last free one today — make it count tomorrow, or type /upgrade for unlimited.