Equations of Tangents Perpendicular to Given Curves

In this article, we will explore the process of finding the equations of tangents to given curves that are perpendicular to the tangents at specific points. This involves a series of differentiation and algebraic steps, providing a deep dive into the intersection of calculus and geometry.

Given Curves and Tangent Conditions

Let's start with two given curves and the conditions to find the tangent equations that are perpendicular to them.

First Curve

The first curve is defined by the equation:

y x^2 - 4x^2

First, we differentiate this equation with respect to x to find the slope of the tangent line:

(frac{dy}{dx} 2x - 4)

We need to find the slope of the tangent to this curve at the point (1, -1). Substituting x 1 into the derivative:

(frac{dy}{dx}bigg|_{x1} 2(1) - 4 -2)

Now, we need to find a tangent to the second curve such that it is perpendicular to the tangent at the point (1, -1). The slope of a line perpendicular to another is the negative reciprocal of that slope. Therefore, the slope of the required tangent is:

(frac{1}{2})

Second Curve

The second curve is defined by the equation:

y frac{x-1}{x^2})

We differentiate this equation with respect to x to find the slope of the tangent line:

(frac{dy}{dx} frac{2}{x^2})

Since the required tangent is perpendicular to the tangent at (1, -1), its slope must be the negative reciprocal of -2, which is (frac{1}{2}). Therefore, we set the derivative of the second curve equal to (frac{1}{2}):

(frac{2}{x^2} frac{1}{2})

From this equation, we solve for x:

(x^2 4)

So, x ±2). Now, we find the corresponding y-values for these x-values:

For x 2), y frac{2-1}{2^2} frac{1}{4}) For x -2), y frac{-2-1}{(-2)^2} frac{-3}{4})

These give us two points for the second curve: (2, 1/4) and (-2, -3/4).

Equation of the Tangents

Next, we find the equation of the tangents to the second curve at these points, with the slope being (frac{1}{2}).

At the point (2, 1/4):

The equation of the tangent is of the form:

y - frac{1}{4} frac{1}{2}(x - 2))

Simplifying:

y - frac{1}{4} frac{1}{2}x - 1)

2y - frac{1}{2} x - 2)

2y x - 2 frac{1}{2})

2y x - 1.5)

This simplifies to:

x - 2y 1)

At the point (-2, -3/4):

The equation of the tangent is of the form:

y frac{3}{4} frac{1}{2}(x 2))

Simplifying:

y frac{3}{4} frac{1}{2}x 1)

2y frac{3}{2} x 2)

2y x 2 - frac{3}{2})

2y x - 0.5)

This simplifies to:

x - 2y -7)

Summary: The equations of the tangents to the second curve, which are perpendicular to the tangent at (1, -1) of the first curve, are x - 2y 1) and x - 2y -7).

This detailed analysis showcases the interplay between the slopes of tangents and their perpendicularity, providing a clear path for understanding and solving such problems.