P and Q are two points on the parabola y^2=4 p x, the coordinates of P are (a, b), and the slope

P and Q are two points on the parabola y^2=4 p x, the...

Key Concepts

Elimination of Parameters

In many algebraic problems, parameters present in the equations of curves or lines can be eliminated through substitution using given conditions. This process helps in expressing the final answer in a form that is independent of specific parameters, highlighting underlying invariant properties of the geometric configuration.

Slope of a Line

The slope of a line is the ratio of the vertical change to the horizontal change between any two points on the line. It is a measure of the line's steepness and is calculated as (change in y)/(change in x). In problems involving known slopes, the relationship between the coordinates of points on a line can be exploited to simplify expressions.

Midpoint of a Line Segment

The midpoint of a line segment between two points in a coordinate plane is found by taking the average of their x-coordinates and the average of their y-coordinates. This concept is fundamental in coordinate geometry for determining intermediate points and is particularly useful when dealing with points on curves.

Parabola

A parabola is a conic section that can be defined by a quadratic equation. The standard equation y² = 4px represents a parabola that opens to the right, where the parameter p is related to the distance between the vertex and the focus. Understanding the general form and properties of a parabola is essential in problems where points lie on the curve.

Transcript

00:01 However, there is a question which is that p and q are two points on the parabola y square equal to 4 p x let us draw the parabola to get more clarity so let us say this parabola is y square equal to 4 p x the coordinates of coordinates of p coordinates of p are a comma b and the slope of p q is m so let us say p is here a comma b and let us say q it's okay say if it is q so if this is a comma b and q is let us say c comma and slope of pq is m if i draw pq like this is straight line slope of pq is m now we have two points we put two points and its slope is equal to m that means from formula d minus b divided by c minus a this will be the slope now these are two points both the points are on parabola y square equal to 4 px so much satisfy this so let us plug in a comma b there so b squared equal to 4 p a and d square equal to 4 p c so if we subtract this equation number one this is equation number two and equation number three now satisfying as subtracting equation number three from equation number two it will be b square minus d square for p a minus c so i can write b plus d into b minus d equal to 4p into a minus c so d minus b divide by i will be just dividing it so it will become b plus d okay or d minus b divided by c minus a equal to 4 p by b plus d 4 p by b plus d so if we plug in over here it will become slope equal to 4 p by b plus d 4 p by b plus d equation number 3 equation number four now i have to find the coordinates of midpoint of pq so midpoint of pq that is this point its coordinate will be c plus a by 2 and b plus d by c plus a by 2 and b plus d by 2 will be its coordinate so from here b plus d will be 4 p by m b plus d is 4 p by m and second is c plus a so c plus a no worries y coordinate of midpoint of pq so only y coordinate is needed so b plus d by 2 will be the y coordinate that means the y coordinate of m will be equal to b plus d by 2 so 4 p by 2 this must be the answer...

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