 Hi and welcome to the session. My name is Priyanka and the question says find the equation of the tangent to the curve y is equal to under root 3x minus 2 which is Parallel to the line 4x minus 2y plus 5 is equal to 0 Since the tangent is parallel to this line that means slope of tangent will be equal to slope of this line so let us Firstly find out the slope of the tangent that will be by writing the equation of the curve which is given to us once again Differentiating it with respect to x With respect to x we get dy by dx is equal to 1 multiplied by 3 upon 2 under root 3x minus 2 that implies dy by dx is equal to 3 by 2 Under root 3x minus 2 right now dy by dx at point x and y is Given to us, sorry found out to be this so that means slope of tangent equal to 3 upon under root 3x minus 2 now we have the slope for the tangent We will be finding the slope of the line and then equating them to get the required answer Now here equation of the line is given to us as 4x minus 2y plus 5 is equal to 0 this implies 2y is equal to 4x plus 5 isn't it this further implies y is equal to 2x plus 5 by 2 Now the slope of this line will be equal to 2 right now since the tangent is parallel to the line that means slope of tangent is equal to slope of line we have slope of the tangent as 3 upon 2 into under root 3x minus 2 it will be equal to 2 now on squaring both sides we get in order to Make this under root sign Simplified we need to square both the sides. This gives us 9 upon 4 into 3x minus 2 equal to 4 This further implies 3x minus 2 is equal to 9 upon 16 That is 3x is equal to 9 upon 16 plus 2 That is 9 plus 16 into 2 gives us 32 upon 16 that is after taking the L-sine and We have 3x equal to 41 upon 16 and Finally the value of x as 41 upon 16 into 3 that gives us 48 Now we know that x1 y1 Lies on y equal to under root 3x minus 2 So we can say that therefore y1 will be equal to under root 3x1 minus 2 That implies y1 is equal to we have x1 the value of x1 as 41 by 48 and substituting its value. Let us find the value of y1 We have y1 equal to under root 41 upon 16 minus 2 41 minus 32 upon 16 gives us under root 9 upon 16 Which gives us the value of y1 either to be plus 9 by 16 or minus sorry plus 3 by 4 or minus 3 by 4 hence the point is If we take the positive side of it 41 by 48 comma 3 by 4 now the required equation of the tangent at this point y minus y1 which is 3 by 4 equal to slope of the tangent which we found out above as 2 as it was equal to the slope of the line which we found out to be 2 bracket x minus x1 which is 41 by 48 What we need to do is we need to just simplify it To get the required answer. We have y upon 2 minus 3 upon 8 equal to x minus 41 by 48 this further implies Y upon 2 minus x equal to minus 41 by 8 plus 3 by 8 That is y minus 2x upon 2 is equal to taking 48 as the LCM we have minus 41 plus 18 Y minus 2x upon 2 equal to 41 minus 18 will give us 23 and since negative sign is with a greater number We will write minus 23 upon 48 simplifying it now we have and on cross multiplying it is 24 y minus 48 x equal to minus 23 or on rewriting it we have 48 x minus 24 y equal to 23 this is the required equation of the Tangent that will form our answer to the given question This is 48 x minus 24 y equal to 23 Right this completes the session. Hope you understood the whole concept well to take care of your calculations and have a very nice day ahead