Question

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.

Answer 1

The equation of the given curve is . y=x²-2x+7
On differentiating with respect to x, we get:

(a) The equation of the line is 2x − y + 9 = 0. 

2x − y + 9 = 0 ∴ y = 2x + 9

This is of the form y = mx + c.
∴Slope of the line = 2
If a tangent is parallel to the line 2x − y + 9 = 0, then the slope of the tangent is equal to the slope of the line. Therefore, we have:

2 = 2x − 2

Thus, the equation of the tangent passing through (2, 7) is given by,

Hence, 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) The equation of the line is 5y − 15x = 13.

Hence, the equation of the tangent line to the given curve (which is perpendicular to line 5y − 15x = 13) is

36y+12x-227=0