How to find equations of both lines that are tangent to the curve y = 1 + x^2 and are parallel to the line 12x - y=1?

Parallel lines have an equal slope. As the equation of the
required tangents to the curve are parallel to the line 12x - y = 1, the slope of the
tangents is equal to the slope of 12x - y = 1.

12x - y =
1

=> y = 12x - 1

This
is the equation of the line in the slope-intercept form with the slope given by
12.

The value of the first derivative at any point on the
curve is equal to the slope of the tangent at that
point.

For the curve y = 1 + x^2, y' =
2x

2x = 12

=> x =
6

For x = 6, y = 1 + 6^2 =
37

The required tangent passes through (6, 37) and has a
slope 12.

(y - 37)/(x - 6) =
12

= y - 37 = 12x -
72

=> 12x - y - 35 =
0

There is only one tangent to the curve that is parallel
to the line 12x - y = 1.

The equation of the
tangent is : 12x - y - 35 = 0