Learn Coordinate geometry with Free Lessons & Tips

To find a point on the y-axis that is equidistant from the points (-5, 2) and (9, -2), we first find the midpoint of the line segment connecting these two points. Then, we find the point on the y-axis that has the same y-coordinate as the midpoint.

If three points are collinear, they lie on the same straight line. To find the relation between xx and yy when the points (2,1), (x,y), and (7,5) are collinear, we can use the slope formula.

To find the length of the median through vertex

If points A(4,3) and B(x,5) are on the circle with center O(2,3) it means that the distance between A (or B) and O is equal to the radius of the circle.

To find the value of k, we can use the formula for the area of a triangle formed by three points

Relationship between backward induction and other solutions

If you have gone through the exercises for the previous three sections, you will have seen that the backwards-induction solutions are also Nash equilibria in all those games. It is always true, as stated in the following theorem.

Theorem 3.4.1 Every backwards-induction solution of a perfect-information game is a Nash equilibrium of the associated strategic form. In some games, the set of backwards-induction solutions coincides with the set of Nash equilibria (see, for example, Exercise 3.9). Still, the set of Nash equilibria is typically larger than (is a proper superset of) the set of backwards-induction solutions. Nash equilibria that are not backwards-induction solutions often involve incredible threats.

To see this, consider the following game. The industry is currently a monopoly, and the incumbent monopolist is making a profit of $5 million. A potential entrant is considering whether or not to enter this industry. - If she does not enter, she will make $1 million in an alternative investment. If she does enter, then the incumbent can either fight entry with a price war whose outcome is that both firms make zero profits or accommodate entry by sharing the market with the entrant, in which case both firms make a profit of $2 million. This situation is illustrated in Figure 3.10 with the associated strategic form. We assume that each player is selfish and greedy; that is, it cares only about its profit and prefers more money to less. The backwards-induction solution is (in, accommodate), and it is also a Nash equilibrium. However, there is another Nash equilibrium, namely (out, fight). The latter should be discarded as a “rational solution” because it involves an incredible threat on the incumbent part, namely that it will fight entry if the potential entrant enters. - It is true that if the potential entrant believes the incumbent’s threat, then she is better off staying out. However, she should ignore the incumbent’s threat because she should realize that – when faced with the fait accompli of entry – the incumbent would not want to carry out the threat. 

Section formula :

Friends, in this video, we gonna learn about the section formula of coordinate geometry chapter.