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1. To determine the equation of the normal line to the curve defined by the equation \(2x^3 + 3xy + \frac{2}{y} = 0\) at the specific point \((1, -1)\), one must first compute the derivative of the curve. This involves implicit differentiation, where the relationship between \(x\) and \(y\) is analysed to find the slope of the tangent line at the given point. By differentiating both sides of the equation with respect to \(x\), one can isolate \(\frac{dy}{dx}\) to ascertain the slope of the tangent line at the coordinates in question.
2. Once the slope of the tangent line is established, the next step is to determine the slope of the normal line. The normal line is perpendicular to the tangent line, which means its slope will be the negative reciprocal of the tangent slope. After calculating this value, one can utilise the point-slope form of a linear equation to formulate the equation of the normal line. This process is essential in A Level Maths May Revision, as it reinforces the understanding of derivatives and their applications in geometry.
3. Finally, substituting the coordinates of the point \((1, -1)\) into the point-slope equation will yield the specific equation of the normal line. This equation will provide a clear representation of the line that is perpendicular to the curve at the designated point, illustrating the relationship between the curve and its normals. The entire procedure not only enhances one’s comprehension of calculus concepts but also emphasises the importance of analytical skills in solving complex mathematical problems.
