 Hello friends and how are you all doing today? My name is Priyanka and the question says find the equation of the tangent line to the curve y is equal to x square minus 2x plus 7 which is e parallel to the line 2x minus y plus 9 is equal to 0 b perpendicular to the line 5 by minus 15x is equal to 13. Now let's proceed with the solution. We are given y equal to x square minus 2x plus 7. Now on differentiating both the sides with respect to x we get dy by dx is equal to 2x minus 2 which is equal to slope of the tangent also. So therefore slope of the tangent is equal to 2x minus 2. Now according to the e part we are given that the tangent is parallel to line 2x minus y plus 9 equal to 0. So the slope of this line will be equal to 2x plus 9 where m is equal to 2. Let this be the first equation and this be the second equation. Now since tangent is parallel to the line therefore the slopes remain the same. So on equating 1 and 2 we get 2x minus 2 is equal to 2 which implies x is equal to now when x is equal to 2 then the value of y is x square so that would be 2 square minus y that is minus sorry it was 2x plus 7 which is 4 minus 4 plus 7 which is equal to 7. So hence equation of the tangent passing through 0.27 and slope 2 is y minus 7 that is y1 equal to slope x minus x1 that is 2 which implies y minus 7 is equal to 2x minus 4 which is further equal to 2x minus y plus 3 equal to 0 or we can write it as y minus 2x minus 3 is equal to 0. Now this is the answer for the e part. Let us proceed to the b part. Here the tangent is perpendicular to the line 5y minus 15x equal to 13 which implies we can reform it and write it as 5y is equal to 15x plus 13 which implies y is equal to 15 divided by 5 gives us 3x plus 13 divided by 5 gives us 13 by 5 so the slope of this line is 3. Now since tangent is perpendicular to the line therefore the product therefore product of slopes should be equal to minus 1 that implies 3 into slope of the tangent that was found above as 2x minus 2 is equal to minus 1 this implies 6x minus 6 is equal to minus 1 or 6x is equal to 5 or x is equal to 5 by 6. Now when x is equal to 5 by 6 the value of y is x square that is 5 by 6 the whole square minus 2x plus 7 that implies 25 by 36 minus 5 by 3 plus 7 on taking the L same and simplifying we have 36 as the L sim so we have 25 minus 60 plus 252 giving us the value of y as 217 by 36 so therefore the point is 5 by 6 comma 217 by 36. Now equation of the line passing through point 5 comma 6 comma 2 17 upon 36 and slope that will be finding the value of dy by dx at x equal to 5 by 6 is equal to 2x minus 2 that will be 5 by 3 minus 2 that will be 5 minus 6 by 3 that is equal to minus 1 by 3 right so we have y minus y1 equal to slope that is minus 1 by 3 x minus x1 on simplifying it y minus 217 by 36 is equal to minus x by 3 plus 5 by 18 further we have 36 y minus 217 upon 36 equal to 18 minus 6x plus 5 further simplifying we have 36 y minus 217 equal to minus 12x plus 10 which is 36 y plus 12x minus 227 equal to 0 now this is the answer to the second part so let me write the answers for you once again the answer for the e and the b part are y minus 2x minus 3 equal to 0 and 36 y plus 12x minus 227 equal to 0 respectively hope you understood the whole process well when a tangent is parallel to a line then the slopes of the tangent and the line are equal to each other and when the tangent is perpendicular to the line then the product of the two slopes should be equal to minus 1 just remember all these facts and hope you enjoyed it