The line that passes through the focus of a parabola and is...

The line that passes through the focus of a parabola and is parallel to the directrix intersects the parabola at two points $A$ and $B$ The line segment $\overline{A B}$ is called the latus rectum of the parabola.We work with the parabola $x^{2}=4 c y, c>0$ By $\Omega$ we mean the region bounded below by the parabola and above by the latus rectum. What is the slope of the parabola at the endpoints of the latus rectum?

The line that passes through the focus of a parabola and is parallel to the directrix intersects the parabola at two points $A$ and $B$ The line segment $\overline{A B}$ is called the latus rectum of the parabola.We work with the parabola $x^{2}=4 c y, c>0$ By $\Omega$ we mean the region bounded below by the parabola and above by the latus rectum.
What is the slope of the parabola at the endpoints of the latus rectum?

Key Concepts

Parabola

A parabola is a conic section that can be defined as the set of all points equidistant from a fixed point called the focus and a fixed line known as the directrix. Its standard equations vary based on orientation, and understanding its geometric properties is key in analyzing its curves and tangents.

Focus and Directrix

The focus is a specific point used in the definition and construction of a parabola, and the directrix is a line. Every point on the parabola is equidistant from the focus and the directrix. This relationship is central in establishing the conic's reflective properties and is used to derive its equation.

Latus Rectum

The latus rectum of a parabola is the line segment that passes through the focus and is parallel to the directrix. It represents the width of the parabola at the level of the focus and is often used to measure the ‘openness’ or spread of the parabola.

Derivative (Tangent Slope)

Differentiation is a calculus tool used to compute the slope of the tangent to a curve at a given point. In the context of a parabola, finding the derivative of its equation allows us to evaluate the slope of the curve at specific points, such as those at the endpoints of the latus rectum, which is crucial for understanding the behavior of the parabola at these 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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