45–50. Tangent lines Carry out the following steps. <IMAGE>
a. Verify that the given point lies on the curve.
(x²+y²)²=25/4 xy²; (1, 2)​​

Verified step by step guidance

1. First, substitute the given point (1, 2) into the equation of the curve \((x^2 + y^2)^2 = \frac{25}{4}xy^2\).
2. Calculate \(x^2 + y^2\) by substituting \(x = 1\) and \(y = 2\) into the expression, resulting in \(1^2 + 2^2 = 1 + 4 = 5\).
3. Substitute \(x = 1\) and \(y = 2\) into the right side of the equation \(\frac{25}{4}xy^2\), which becomes \(\frac{25}{4} \times 1 \times 2^2 = \frac{25}{4} \times 4 = 25\).
4. Now, substitute the calculated values into the original equation: \((5)^2 = 25\) and \(25 = 25\).
5. Since both sides of the equation are equal, the point (1, 2) lies on the curve.

Key Concepts

Implicit Differentiation

Implicit differentiation is a technique used to differentiate equations where the dependent and independent variables are not explicitly separated. In this case, we differentiate both sides of the equation (x² + y²)² = 25/4 xy² with respect to x, treating y as a function of x. This allows us to find the slope of the tangent line at a specific point on the curve.

Tangent Line

A tangent line to a curve at a given point is a straight line that touches the curve at that point and has the same slope as the curve at that point. The slope of the tangent line can be found using the derivative of the function at that point. The equation of the tangent line can then be expressed in point-slope form, using the coordinates of the point and the calculated slope.

Verifying Points on Curves

To verify that a point lies on a curve defined by an equation, we substitute the coordinates of the point into the equation. If the left-hand side equals the right-hand side after substitution, the point is confirmed to be on the curve. In this case, substituting (1, 2) into the equation (x² + y²)² = 25/4 xy² will determine if the point lies on the curve.