 Hello and welcome to the session. Let us discuss the following question. It says a man running a race course notes that the sum of the distances from the two flag posts from him is always 10 meter and The distance between the flag posts is 8 meter find the equation of the posts traced by the man To solve this question. We need to know some basic definitions First we start with the definition of ellipse It says that an ellipse is the set of all points in the plane the sum of whose Distances from two fixed points in a plane is constant Now we relate this definition with this question It says a man running a race course notes that sum of distances from two flag posts from him is always 10 meter That means while running he traces a path where on all points the sum of the distances from two fixed points is constant and Here that constant distance is given to be 10 meter That means the man is Tracing an ellipse while running and The two points from where the sum of the distances is constant are called foci of the ellipse So man is tracing an ellipse while running and the flag posts are f1 and f2 and from the point P the sum of the distances That is P f1 plus P f2 is 10 meter and f1 Is the focus f2 is also focus and distance between f1 and f2 is? 8 meter F1 and f2 are two foci of the ellipse and f1 has the coordinate minus C0 and f2 has coordinates C0 Now we are given the distance between f1 and f2 is 8 meter so o f1 is Equal to o f2 is half of f1 f2 that is 4 meter Because distance between f1 f2 is 8 meter which is equal to 2 C right So C is 4 meter Now let the man is At the point P Then in an ellipse sum of the distances from two fixed points from a certain point in The plane so here we are given that P f1 plus P f2 is Equal to the major axis P f1 plus P f2 is Equal to 2 a Now P f1 plus P f2 is 10 meter So this implies a is 5 Now since the man is tracing an elliptical path We have to find the equation of this ellipse for that we need to Find the semi minor axis because equation of the ellipse is x squared upon a squared plus y squared upon b squared is equal to 1 where a is the semi major axis and b is the semi minor axis Now here we see that C is the distance of focus from the center now we know that Square of the semi major axis is equal to the square of semi minor axis Plus the square of the distance of the focus from the center Now a is 5 C is 4 so substituting the values 5 square is equal to b squared plus 4 square and This implies b square is equal to 25 minus 16 and this implies B square is equal to 9 and this implies b is equal to plus minus 3 Hence the equation of the posts Traced by the man is given by x squared upon a squared where a is 5 plus Y squared upon b squared where b is plus minus 3 the square of plus minus 3 is equal to 1 which is equal to x squared upon 25 plus Y squared upon 9 is equal to 1 Hence the equation of electrical path is x squared upon 25 plus Y squared upon 9 is equal to 1 and this completes the Question and the session bye for now take care. Have a good day