 Hello and welcome to the session. Let us discuss the following question. It says, if PAB is seeking to a circle, intercepting it at A and B and PT is tangent, prove that PA into PB is PT squared. Using the above result, solve the following. Find the value of AB when PT is 5 centimeter and PA is 4 centimeter. So, let us now move on to the solution. We'll first prove this theorem that PA into PB is PT squared, where PAB is the secant to the circle. So, let us first write what is given to us. We are given a secant PAB to a circle with center O, intersecting it at A and B and tangent PT to a circle to the circle and what we have to prove? We have to prove that PA into PB is PT squared and to prove this we do the following construction. Join O to the midpoint M of AB and also to A, B and a tangent PT to a circle. We have to join O with A only and then we have to join PNT. Let us now start the proof equal to PM minus AM and also PB is equal to PM plus MB but M is the midpoint of AB. Therefore, AM is equal to MB. So, we have PB is equal to PM plus AM. This is because AM is equal to MB. So, we have PA into PB is equal to PM minus AM into PM plus AM. So, this is equal to PM square minus AM square using the formula of A minus B into A plus B is A square minus B square. Now, again OM is perpendicular to AB. Therefore, OMP is the right angles triangle. Therefore, by the Pythagoras theorem is equal to OP square minus OM square by Pythagoras theorem. Right? And also AM square is equal to OA square M square is equal to OA square minus OM square again by using Pythagoras theorem. So, thus PA into PB is equal to square minus OM square minus OA square minus OM square. This is because PA into PB is PM square minus AM square but PM square is OP square minus OM square and AM square is OA square minus OA square minus OM square. So, we have OP square minus OM square minus OA square plus OM square since we open the bracket and we have negative sign outside the bracket. So, sign will change and OM square gets cancelled with OM square. We have OP square minus OA square that is OP square minus OT square and this is because OA is equal to OT being the radius of the circle and OP square minus OT square is equal to PT square. This is why again using Pythagoras theorem. This is because PTO is a right angle triangle. So, PT square is equal to OP square minus OT square that is by using Pythagoras theorem. Now, the next part says, find the value of AB when PT is 5 centimeter and PA is 4 centimeter. Now, we have PT is 5 centimeter, PA is 4 centimeter, we have to find AB, right? Now, from the above result, we have PA into PB is PT square. So, we have 4 into PB is equal to PT square that is 5 square. So, this implies PB is equal to 25 by 4. So, we have PB is equal to 25 by 4. Now, we have to find the length of AB. Now, AB is equal to PB minus PA. Now, PB is 25 by 4 and PA is 4 centimeter. Now, taking LCM, we have 25 minus 16 upon 4. Now, 25 minus 16 is 9 upon 4. Therefore, AB is equal to 9 by 4 centimeter. So, this completes the question and the session. Bye for now. Take care. Have a good day.