Find the equation of the tangent line to the curve *y* = *x*^{2} − 2*x* + 7 which is

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

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

#### Solution

`The equation of the given curve is *y* = *x*^{2} − 2*x* + 7

On differentiating with respect to *x*, we get:

`dy/dx = 2x - 2`

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

2*x* − *y* + 9 = 0 ⇒ *y* = 2*x *+ 9

This is of the form *y* = *mx *+ *c*.

∴Slope of the line = 2

If a tangent is parallel to the line 2*x* − *y* + 9 = 0, then the slope of the tangent is equal to the slope of the line.

Therefore, we have:

2 = 2*x* − 2

(b) The equation of the line is 5*y* − 15*x* = 13.

5*y* − 15*x* = 13 ⇒ `y = 3x + 13/5`

This is of the form *y* = *mx *+ *c*.

∴Slope of the line = 3

If a tangent is perpendicular to the line 5*y* − 15*x* = 13, then the slope of the tangent is