Question

Find the equation of the normals to the curve y = x³ + 2x + 6 which are parallel to the line x + 14y + 4 = 0.

Answer 1

The equation of the given curve is y = x³ + 2x + 6.

The slope of the tangent to the given curve at any point (x, y) is given by,

∴ Slope of the normal to the given curve at any point (x, y) =

The equation of the given line is x + 14y + 4 = 0.

x + 14y + 4 = 0 ⇒ (which is of the form y = mx + c)

∴Slope of the given line =

If the normal is parallel to the line, then we must have the slope of the normal being equal to the slope of the line.

When x = 2, y = 8 + 4 + 6 = 18.

When x = −2, y = − 8 − 4 + 6 = −6.

Therefore, there are two normals to the given curve with slopeand passing through the points (2, 18) and (−2, −6).

Thus, the equation of the normal through (2, 18) is given by,

And, the equation of the normal through (−2, −6) is given by,

Hence, the equations of the normals to the given curve (which are parallel to the given line) are