 Hello and how are you all today? My name is Priyanka and the question says find the equation of all lines having slope minus 1 that are tangents To the curve y is equal to 1 upon x minus 1 where x is not equal to 1 now here We are given the equation of the curve as y is equal to 1 upon x minus 1 and since we are given slope as minus 1 that means first of all we will be finding dy by dx that will be by differentiating y with respect to x and on doing so we get dy by dx equal to on applying the quotient rule we have it as minus 1 upon x minus 1 the whole square right Also, we are given That dy by dx That is slope is equal to minus 1 right so on equating it we have Minus 1 is equal to minus 1 upon x minus 1 the whole square and Simplifying it further We have x minus 1 the whole square equal to 1 that further gives us x minus 1 is equal to under root 1 x minus 1 is equal to plus minus 1 or We can say that the value of x is equal to 2 or it is equal to 0 now when the value of x is Equal to 2 then the value of y comes out to be 1 upon x minus 1 That is 2 minus 1 which is further equal to 1 so the point is 2 comma 1 now when The value of x is 0 the value of y will be 1 upon 0 minus 1 that gives us the answer as minus 1 so the point is 0 minus 1 right now the equation of the line with slope minus 1 is passing through the point to 1 Why minus y 1 equal to n that is the slope bracket x minus x 1 now we have y 1 as 1 slope as minus 1 and x sorry x 1 as 2 So on simplifying we have the value of Or sorry, we have the equation as we're simplifying it x plus y is equal to 3 and The equation of the line with slope minus 1 and passing through point 0 minus 1 is by minus y 1 that will be plus 1 now Equal to minus 1 that is the slope x minus x 1 that is 0 Or we have x plus 1 equal to minus x or x plus y plus 1 is equal to 0 right Now here also we can write this equation as x plus y minus 3 equal to 0 we have all the terms in our left-hand side So the answer to the session is The first equation can be any of these two that can be X plus y can be also written as y plus x plus 1 is equal to 0 and the other equation is x plus y that can be Y plus x Minus 3 is equal to 0 so these are The equation of all lines having slope minus 1 that are tangent to the curve y is equal to 1 upon x minus 1 Where x is not equal to 0 this completes our session hope you understood it well and enjoyed it to have a nice day ahead