What is the slope of the parabola at the endpoints of the latus rectum?

Name: Problem 36, Chapter 10: Calculus: One and Several Variables
Uploaded: 2021-11-08T05:37:47-08:00
Duration: 3 min 20 s
Channel: Harmender Singh Yadav
Description: What is the slope of the parabola at the endpoints of the latus rectum?

What is the slope of the parabola at the endpoints of the...

Key Concepts

Parabola

A parabola is a type of conic section that can be defined as the set of all points equidistant from a fixed point called the focus and a fixed line called the directrix. Its geometric and algebraic properties, including axis of symmetry and vertex, form the basis for understanding its shape and behavior under various transformations and operations in both geometry and calculus.

Latus Rectum

The latus rectum of a parabola is a line segment that passes through the focus and is perpendicular to the axis of symmetry. It has endpoints on the parabola and its length is directly related to the parameter defining the parabola’s width. This concept is essential in linking the geometric shape of the parabola with its algebraic expression.

Derivative

The derivative of a function provides the slope of the tangent line to the curve at any given point, representing the instantaneous rate of change. In the context of a parabola, computing the derivative allows us to determine the slope at specific points such as the endpoints of the latus rectum, which can reveal important characteristics of the curve's shape at those points.

Transcript

00:02 In this question, an equation of the parabola is given as x square equals to 4cy.

00:06 We need to find out the slope of the tangent at the end points of the latest rectum of this parabola.

00:16 Now, we drawn the figure for such a situation where the equation of the parabola is having such a parabola which is symmetric about y -axis and having center at 0 -c -c -c -c on the y -axis.

00:30 And this is the latest rectum of this parabola therefore the we know the length of the later spectrum of the parabola we already calculated that will be equal to 4c so the length of the para later set term is equal to 4c therefore the coordinates where it intersect the parabola will be here this will be 2c comma c that is the coordinate of this focus that is y coordinate will be c and x coordinate will be 2c and here for this this will be minus 2c comma c so we got the coordinates of these points where the latest rectum intersect this parabola now we can calculate the the slope of the tangent at these points we know the slope of the tangent can be calculated by d y by d x and we have the equation of this parabola as x square equal to 4 c y from here we will get y equals to x square over 4c and by differentiating it with respect to x we will get d y over d x will be equal to 2x over 4c and now to get the slope of the that is tangent at end points of the latest rectum we will substitute x as 2c so d y y by d x will be the slope of the tangent that is m let us consider these points as a and b here this will be a and this point as b so the slope at a a we are considering it as m a that will be d y by d x at x equals to 2c so this will become two times two c over 4c so this will become 4 c over 4c that will be one therefore we get the slope of the tangent at point a will be 4 c over 4 c that is one and now similarly we can find out the slope at point slope of the tangent at point b we can consider that m b that will be d y by d x and substituting x as it was minus 2c so substituting it as minus 2c we will get 2x over 4c as 2 times minus 2c over 4c this will become minus 4c over 4c that will be minus 1...

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