Find the equation of the tangent line to the curve y = x2 – 2x +7 which is

(a) parallel to the line 2x – y + 9 = 0


(b) perpendicular to the line 5y – 15x = 13.


(a) It is given that equation of the curve is y = x2 – 2x +7


On differentiating with respect to x, we get



The equation of the line is 2x – y + 9 = 0


y = 2x + 9


Slope of the line = 2


Now we know that if a tangent is parallel to the line 2x – y + 9 = 0, then


Slope of the tangent = Slope of the line


2 = 2x – 2


2x = 4


x = 2


Now, putting x = 2, we get


y =4 -4 + 7 = 7


Then, the equation of the tangent passing through (2,7)


y – 7 = 2(x – 2)


y – 2x – 3 = 0


Therefore, the equation of the tangent line to the given curve which is parallel to line 2x – y + 9 = 0 is y – 2x – 3 = 0.


(b) It is given that equation of the curve is y = x2 – 2x +7


On differentiating with respect to x, we get



The equation of the line is 5y – 15x = 13


y =


Slope of the line = 3


Now we know that if a tangent is perpendicular to the line 5y – 15x = 13, then



2x – 2=


2x =


x =


Now, putting x = , we get


y =


Then, the equation of the tangent passing through


y – = (x – )



36y – 217 = -2(6x -5)


36y+12x – 227 = 0


Therefore, the equation of the tangent line to the given curve which is perpendicular to line 5y – 15x = 13 is 36y+12x – 227 = 0.


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