 Hello and welcome to the session. In this session we are going to discuss how to sketch the ellipse. The standard equation of the curve that is ellipse is given by x square by a square plus y square by b square is equal to 1. Now if we put the coordinates 00 in this equation we see that 00 does not satisfy the equation of the ellipse that is x square by a square plus y square by b square is equal to 1. Therefore we can say that the curve does not pass through the origin. Now we will see the intersection of the curve with x axis. The curve will intersect the x axis when y is equal to 0. So let us put y is equal to 0 in this equation that is x square by a square plus y square by b square is equal to 1. Therefore we get x square by a square is equal to 1 which implies that x square is equal to a square which implies that x is equal to plus minus of a. The curve will intersect x axis point a with the coordinates a0 and point a dash with the coordinates minus a0. Now you have seen that the curve will intersect the x axis at two points that is point a with coordinates a0 and point a dash with coordinates minus a0. Now we will see the intersection of the curve with y axis. The curve will intersect the y axis when x is equal to 0. So let us put x is equal to 0 in the equation x square by a square plus y square by b square is equal to 1. Therefore we get y square by b square is equal to 1 which implies that y square is equal to b square which further implies that y is equal to plus minus of b. So the curve will intersect the y axis at the point b with the coordinates 0 b and point b dash with the coordinates 0 minus b. That is the curve will intersect the y axis at two points b with the coordinates 0 b and b dash with the coordinates 0 minus b. Now as b is less than a therefore distance from center say o to b and b dash will be less than distance from center o from a and a dash that is o b and o b dash will be less than o a and o a dash. Now we will check the symmetry of the curve about x axis and y axis. As we are given the equation x square by a square plus y square by b square is equal to 1 and it can be rewritten as y is equal to plus minus b into square root of 1 minus x square by a square or we can also write it as x is equal to plus minus a into square root of 1 minus y square upon b square. Now for y is equal to plus minus b into square root of 1 minus x square by a square we can conclude that for each value of x where minus a is less than x is less than a y we always have two equal and opposite values that is the curve is symmetrical about x axis also if you change the value of y into minus y in the equation x square by a square plus y square by b square is equal to 1 then the equation remains unchanged as there is a square of y and square of a negative and a positive number are seen. Therefore we can conclude that the curve is symmetrical about x axis. Now from x is equal to plus minus a into square root of 1 minus y square by b square we can conclude that for each value of y where minus b is less than y is less than b x will have two equal and opposite values that is the curve is symmetrical about y axis also if we change the value of x into minus x in the equation x square by a square plus y square by b square is equal to 1 it remains unchanged. So we can conclude that the curve is symmetrical about y axis. Now we will find out imaginary values for both x and y. Now the given curve x square by a square plus y square by b square is equal to 1 can be rewritten as x square is equal to plus minus a upon b into square root of b square minus y square. Now x will be imaginary if b square minus y square is less than 0 that is b square is less than y square we can say when y is greater than b and y is less than minus b. Therefore no part of the curve will lie above point b with coordinates 0 b or below point b dash with coordinates 0 minus b. As we know that y can be written as plus minus b upon a into square root of a square minus x square. So y will be imaginary if a square minus x square is less than 0 that is f is greater than a square or we can write x is greater than a x is less than minus a. Therefore no part of the curve lies to the right of point a with coordinates a 0 to the left of point a dash with coordinates minus a 0. Now when we put x is equal to a in the equation x square by a square plus y square by b square is equal to 1 we get y square by b square is equal to 0. Which implies that y square is equal to 0 when x is equal to a then y square is equal to 0 that is it gives two values of y. Therefore we can say that the line x is equal to a is a tangent to the curve at a as it represents an equation of a line. Similarly when we put y is equal to b in the same equation then we get x square is equal to 0 that is it gives two values of x therefore we can say that the line is equal to b is a tangent to the curve it represents an equation of a line. As we know that the curve square by a square plus y square by b square is equal to 1 is symmetrical about x axis as well as y axis. Therefore x is equal to minus a and y is equal to minus b also represents the tangent to the curve and b dash respectively. The equation of the curve can also be written as x is equal to less minus a upon b into square root of b square minus y square. Now as y increases from minus b to 0 x increases from 0 to a to the right of y axis decreases from 0 to minus a to the left of y axis as y increases from 0 to b x decreases from a to 0 to the right of the y axis increases from minus a to 0 to the left of y axis thus the curve is a closed one. Now we can sketch the graph by using all of the above data here we can see that the curve is symmetrical about x axis and y axis. There is no part of the curve lines above the point b with the coordinates 0 b or below the point b dash with the coordinates 0 minus b. Also no part of the curve lies to the right of point a with coordinates a 0 and to the left of a dash with coordinates minus a 0. Here we can also see as y increases from minus b to 0 x increases from 0 to a to the right of y axis and decreases from 0 to minus a to the left of y axis. Also as y increases from 0 to b x decreases from a to 0 to the right of y axis and increases from minus a to 0 to the left of y axis. Therefore we get a closed curve and this is the required sketch of the ellipse x square by a square plus y square by b square is equal to 1. This completes our session. Hope you enjoyed this session.