 Hello and welcome to the session. My name is Mansi and I am going to help you with the following question. The question says a hollow cone is cut by a plane parallel to the base and the upper portion is removed. If the curved surface of the remainder is 8 by 9 of the curved surface of the whole cone, find the ratio of the line segments into which the cone's altitude is divided by the plane. So by the interpretation of the question we have, the cone is cut off by this line that is parallel to the base. So let us start with the solution to this question. First of all, let the heights of the cone's A, D, E and A, B, C be H1 and H2. That means let this be H1 and this be H2. Now in similar triangles A, D, O dash and A, B, O. So first of all we see that in triangle A, D, O dash and triangle A, B, O. The first thing that we have is angle A, O dash, D is equal to angle A, O, B because each of them is equal to 90 degree. Also we have angle A, D, O dash is equal to angle A, B, O because these two act as parallel lines and this acts as a transversal. So they become the pair of alternate angles. So due to these two reasons we have triangle A, D, O dash is similar to triangle A, B, O. Since these two triangles are similar therefore we will have the corresponding sides are proportional that means A, D by A, B will be equal to A, O dash by A, O that is equal to A, O dash is height of the smaller cone and A, O is the height of the bigger cone that is H1 by H2. So we have D, O dash by B, O will also be equal to H1 by H2. So D, O dash by B, O will also be equal to H1 by H2. Now we see that curved surface area of cone A, D, O dash is equal to pi RL where R is the radius L is the slant height that is equal to 22 by 7 into R is DO dash and slant height is AD and similarly we see that curved surface area of the cone ABO. We see that here we have curved surface area of the cone ADE is equal to pi RL that is equal to 22 by 7 into DO dash into AD. Similarly curved surface area of the cone ABC is equal to 22 by 7 into BO into AB that is again pi RL since curved surface area of remainder is 8 by 9 of the curved surface area of the cone ABC. So curved surface area of cone ADE is the curved surface area of cone ABC or of the cone ADE is 1 minus 8 by 9 that is 1 by 9th of surface area of cone ABC. Therefore we can say that surface area of cone is equal to 1 by 9 into 22 by 7 into BO into AB because we see that this is the surface area of cone ABC. This implies 22 by 7 into DO dash into AD is equal to 1 by 9 into 22 by 7 into BO into AB that is DO dash divided by BO is equal to AB divided by AD into 1 by 9 because 22 by 7 gets cancelled with 22 by 7. This implies H1 by H2 is equal to H2 by H1 into 9 because in earlier part of the question we have proved that DO dash by BO is equal to H1 by H2 and AB by AD was equal to H2 by H1. So we can say that on cross multiplication H1 square divided by H2 square is equal to 1 by 9 taking square root on both the sides we have H1 by H2 is equal to under the root 1 by 9 that is equal to plus minus 1 by 3. Now since the ratio of two heights cannot be a negative quantity therefore we can say that H1 by H2 is equal to 1 by 3. Now on taking reciprocal on both the sides we have H2 by H1 is equal to 3 by 1 or H2 by H1 minus 1 equal to 3 minus 1 subtracting 1 from both the sides this gives us H2 minus H1 divided by H1 is equal to 2 again taking reciprocal we have H1 divided by H2 minus H1 is equal to 1 by 2. So we see that this is the ratio of line segments into which the cones altitude is divided by the plane. So we say that our answer to the question is that the required ratio is 1 is to 2. So this is our answer to the question. I hope that you understood the question and enjoyed the session. Have a good day.