d) Find the equation of the tangent and normal line to the curve 12(x^2+y^2)=25xy at the point (3,4) (3 Marks) e) A body moves in a straight line so that its distance S meters from a

To find the equation of the tangent and normal lines to the curve \(12(x^2 + y^2) = 25xy\) at the point \((3, 4)\), we need to follow these steps:1. **Implicit Differentiation**: Differentiate the given equation with respect to .2. **Evaluate the Derivative at the Given Point**: Find the slope of the tangent line at \((3, 4)\).3. **Equation of the Tangent Line**: Use the point-slope form of the equation of a line.4. **Equation of the Normal Line**: The normal line is perpendicular to the tangent line, so its slope is the negative reciprocal of the tangent line's slope.Let's go through these steps in detail:### Step 1: Implicit DifferentiationGiven: Differentiate both sides with respect to : Rearrange to solve for : ### Step 2: Evaluate the Derivative at the Given PointSubstitute and into the derivative: So, the slope of the tangent line at \((3, 4)\) is .### Step 3: Equation of the Tangent LineUsing the point-slope form of the equation of a line: where is the slope and \((x_1, y_1)\) is the point \((3, 4)\): Simplify: ### Step 4: Equation of the Normal LineThe slope of the normal line is the negative reciprocal of the slope of the tangent line: Using the point-slope form again: Simplify: So, the equations are:- Tangent line: - Normal line: ### SummaryThe equation of the tangent line to the curve \(12(x^2 + y^2) = 25xy\) at the point \((3, 4)\) is: The equation of the normal line to the curve at the same point is: