 Hello and welcome to the session. In this session we will discuss a question which says that the angle of elevation of a cloud from a point 60 meters above the surface of the lake is 30 degrees and the angle of depression of the reflection of the cloud in the lake is 60 degrees. Find the height of the cloud from the surface of the lake. Now before starting the solution of this question we should know some results. First is, inside the ABC, tan theta is equal to perpendicular to base which is here AB over BC and second is the angle of elevation. Now here in this diagram a person is standing at a point A and he is viewing the object that is lying at the point C. So the line AC will become the line of sight. So the angle BAC that is this angle which is formed by the line of sight with the horizontal is called the angle of elevation of the object from the eye of the observer. So here this is angle of elevation is angle PAC and third is the angle of depression. Now here suppose that a person is standing on the top of the tower AB that is on the point A and he is observing an object which is lying at the point C. So the line AC will become the line of sight. So here the angle CAD which is formed by the line of sight with the horizontal is called the angle of depression. So here the angle of depression is angle CAT. So the angle of depression is the angle when the point is below the horizontal level that is the case when we lower our heads to look at the point being viewed. Now these results will work out as a key idea for solving out this question. And now when we start with the solution, now let AB be the surface of the lake AC be the height of cloud from the surface of the lake D which is this point is the reflection of the cloud which is at the point C. So D is the reflection of the cloud C. Now let AC is equal to X meters that is this height of the cloud from the surface of the lake. Therefore as D is the reflection of the cloud C therefore AC will be equal to AD. AC is equal to AD as D is the reflection of C which implies AD is also X meters. Now it is given in the question that the angle of elevation of a cloud from a point 60 meters above the surface of the lake is 30 degrees and the angle of depression of the reflection of the cloud in the lake is 60 degrees. Now let us take a point E above the surface of the lake such that BE is equal to 60 meters. Now using the result of angle of elevation and angle of depression we get the angle of elevation that is angle CEF is equal to 30 degrees and angle of depression that is the angle DEF is equal to 60 degrees. So we have angle CEF is equal to 30 degrees and angle is equal to 60 degrees. Now from the diagram AF is equal to BE which implies as BE is equal to 60 meters so AF will be also 60 meters. Now AC here is X meters 2 meters so therefore putting these values here this will be X meters is equal to what implies FC is equal to X minus 60 meters. Now DEF can be written as now DA is equal to X meters is equal to 60 meters so putting these values here this implies is equal to X meters this is equal to now let this result which is given lucky idea 30 degrees is equal to but the angle over base which is FC over AF which implies time 30 degrees is equal to 1 by root 3 which is equal to which is 2 meters so it will be X minus 60 meters over which we have taken as Y meters. Further implies 1 by root 3 is equal to X minus 60 over Y which further implies on cross multiplying Y is equal to root 3 into X minus 60 the whole meters. Now 60 degrees is equal to perpendicular over base that is DF 60 degrees is root 3 is equal to DF now DF here is X plus 60 meters so it will be X plus 60 meters over as Y meters which implies root 3 is equal to X plus 60 over Y the value of Y here this implies root 3 is equal to X plus 60 into X minus 60 the whole. X multiplying this implies root 3 into root 3 into X minus 60 the whole is equal to X plus 60 which further implies 3 into X minus 60 the whole is equal to X plus 60 this implies 3X minus 180 is equal to X plus 60 which further implies 3X minus 8 is equal to 60 plus 180 equal to 120 now we have taken AC as the height of the cloud from the surface of the lake and AC is taken as X meters therefore is equal to 120 meters that's all for the session hope you all have enjoyed the session