 Hello and welcome to the session. In this session we will discuss a question that says that find the equation of the plane passing through the line of intersection of the planes minus x plus 2y plus z minus 4 is equal to 0 and 4x plus pi y minus 3z plus 5 is equal to 0 and cutting half equal intercepts on the words and o y axis. Now before starting the solution of this question we should know our result. And that is the equation of the plane passing through the intersection a1x, b1y plus c1z plus d1 is equal to 0 and a2x plus b2y plus c2z plus d2 is equal to 0 is given by plus b1y plus c1z plus d1 the whole plus lambda into a2x plus d2y plus c2z plus d2 the whole is equal to 0 where lambda is any arbitrary constant. Now this result will work out as a key idea for solving out this question. And now we will start with the solution. Now in the question we have to find the equation of the plane passing through the line of intersection these planes and cutting off equal intercepts on the words and o y axis. Now using this result which is given in the key idea the equation dash minus 4 is equal to 0 plus 5 is equal to 0 is given by minus x plus dash minus 4 the whole plus lambda into 4x plus 5 y minus 3z plus 5 the whole is equal to 0. This implies lambda the whole into x under the whole into y into z equation number 1 is the plane which is given by equation number 1 cut-off equal intercepts. Now the plane which is given by equation number 1 y is equal to 0 is equal to 0 in equation number 1 it will become 4 lambda the whole if lambda is equal to 0 this 4 lambda the whole into x is equal to 4 minus 5 lambda which further improve the plane which is given by equation number 1 intercept z is equal to 0 therefore equal to 0 z is equal to 0 in equation number 1 it will become lambda the whole into y plus 5 lambda is equal to 0 lambda the whole into y is equal to which further implies y is equal to 4 minus 5 lambda whole upon 2 plus 5 lambda this is the x intercept and this is the y intercept now the intercept minus 1 plus whole lambda is equal to 4 minus 5 lambda whole upon 2 plus 5 lambda now cross multiplying this implies plus 20 lambda minus 10 lambda minus 25 lambda square is equal to 16 lambda minus 20 lambda square minus 4 this further implies minus 5 lambda square minus 11 lambda plus 12 is equal to 0 which further implies 5 lambda square plus 11 lambda minus 12 is equal to 0 this is the quadratic equation in lambda so we can find the value of lambda by using the quadratic formula that is lambda is equal to minus b that is minus 11 b square that will be minus 4 into now this is equal to 240 whole upon 10 which is further equal to minus 11 this is equal to 1 minus 11 19 whole upon lambda is equal to this is equal to this is equal to 4 by 5 lambda is equal to 19 whole upon 10 which is equal to minus 13 by 10 which is equal to minus 3 lambda is equal to 4 by 5 will become as you can see here that by putting lambda is equal to 4 by 5 we have that equal to 0 as 0 over n thing is equal to 0, so this 4 by 5 we will get y into set is equal to 0. And this symbol that is when lambda is equal to 4 by 5, z minus 4 by whole plus z minus 4 minus 12 x minus 15 y plus 9 z minus 15 is equal to 0. Now this implies minus minus 13 then z minus 19 is equal to 0, 13 x plus 13 y which is the required equation of the given equation and that's all for this session. Hope you all have enjoyed this session.