 Hello and welcome to the session. In this session we discussed the following question that says find the equation of the tangent to the curve y is equal to square root of 3x minus 2 which is parallel to the line 4x minus 2y plus 5 equal to 0. Before we move on to the solution, let's see what is the equation of the tangent at the point x naught y naught to the curve y equal to fx. Its equation is given by y minus y naught is equal to dy by dx at x naught y naught into x minus x naught. This is the key idea that we use for this question. Let's move on to the solution now. Now the given curve is y is equal to square root of 3x minus 2. Let this be equation 1. Now let's differentiate this equation 1. Therefore we get dy by dx is equal to 1 upon 2 into 3x minus 2 to the power minus 1 upon 2 into differential of 3x minus 2. Now differential of 3x minus 2 is 3. So this is into 3. So this would be equal to 3 upon 2 into square root of 3x minus 2 that is dy by dx. So this dy by dx is the slope of the tangent to the curve 1. So we have slope of the tangent to the curve 1 given by m1 is equal to 3 upon 2 into square root of 3x minus 2. Next we consider the given line. The equation of the line is 4x minus 2y plus 5 equal to 0. Let this be equation 2. Now again we differentiate both sides with respect to x so as to get the slope of the line. So differentiating both sides with respect to x we get 4 into 1 minus 2 into dy by dx is equal to 0. So from here we get dy by dx is equal to 4 upon 2 that is equal to 2. And this dy by dx is the slope of the given line 2 thus we have slope of the line is given by m2 is equal to 2. In the equation we have that the tangent to the given curve is parallel to the given line. So as the tangent to the curve 1 is parallel to the line 2 therefore slope of the tangent to the curve which is m1 would be equal to slope of the line that is m2. Now m1 is 3 upon 2 into square root of 3x minus 2 is equal to m2 which is 2. From here let's find out x. So we have 4 into square root of 3x minus 2 is equal to 3 or square root of 3x minus 2 is equal to 3 upon 4. Now squaring both sides we get 3x minus 2 is equal to 9 upon 16 or 3x is equal to 9 upon 16 plus 2 that is 3x is equal to 41 upon 16 or you can say we get x is equal to 41 upon 48. This is the value for x. Now as we have got the value for x let's find out the value for y. So substituting x equal to 41 upon 48 in equation 1 we get that is in this equation so we have y is equal to square root of 3 into 41 upon 48 minus 2. Or you can say y is equal to square root of 41 upon 16 minus 2 since 3 16 times is 48. This gives us y is equal to square root of 9 upon 16 that is equal to 3 upon 4. Thus we get y equal to 3 upon 4. So now we have got the value for x and y. Thus we can say that the point on the curve 1 at which the tangent is parallel to line 2 is the point which coordinates x, y. That is 41 upon 48 3 upon 4. So now the equation of the tangent point 41 upon 48 3 upon 4 which is parallel to line 2 is given by the equation y minus y naught that is 3 upon 4 is equal to slope of the tangent which is 2 as we know that the slope of the line as the slope of the tangent would be same since they are parallel to each other. So it would be 2 into x minus x naught which is 41 upon 48. Thus y minus 3 upon 4 is equal to 2x minus 41 upon 24. Further 4 y minus 3 whole upon 4 is equal to 48x minus 41 whole upon 24. Now 4 6 times is 24 so we get 6 into 4 y minus 3 the whole is equal to 48x minus 41. Further we have 24 y minus 18 is equal to 48x minus 41 that is 48x minus 24 y minus 41 plus 18 is equal to 0. Or you can say we get 48x minus 24 y minus 23 is equal to 0. So this is the required equation of the tangent that is the required equation of the tangent is 48x minus 24 y minus 23 is equal to 0. So this is our final answer. This completes the session. Hope you have understood the solution of this question.