In this page tangent and normal question6 we are going to see solution of some practice questions from the worksheet.

(6) Find the equations of normal to y = x³ - 3 x that is parallel to 2 x + 18 y - 9 = 0.

**Solution:**

Here the normal line drawn to the curve is parallel to the line 2 x + 18 y - 9 = 0

Slope of the line parallel to the normal line to the curve

m = - coefficient of x/coefficient of y

m = -2/18

= -1/9 ----- (1)

Slope of the line parallel to the normal line to the curve = -1/9

Slope of the tangent to the curve y = x³ - 3 x

differentiate with respect to "x"

dy/dx = 3 x² - 3 (1)

= 3 x² - 3

slope of the normal line to the curve = -1/(3 x² - 3) ----- (2)

(1) = (2)

-1/9 = -1/(3 x² - 3)

3 x² - 3 = 9

3 x² = 9 + 3

3 x² = 12

x² = 12/3

x² = 4

x = √4

x = ± 2

to find the y-coordinate we have to apply the x values in the equation of the curve not in the given line

x = 2 x = -2

y = x³ - 3 x y = x³ - 3 x

= 2³ - 3 (2) = (-2)³ - 3 (-2)

= 8 - 6 = - 8 + 6

= 2 = -2

point of contact are (2,2) (-2,-2)

Equation of the normal passing through the point (2,2) and slope is -1/9

(y - y₁) = (-1/m) (x - x₁)

(y - 2) = (-1/9) (x - 2)

9 (y - 2) = -1(x - 2)

9 y - 18 = - x + 2

x + 9 y - 18 - 2 = 0

x + 9 y - 20 = 0

Equation of the normal passing through the point (-2,-2) and slope is -1/9

(y - y₁) = (-1/m) (x - x₁)

[y - (-2)] = (-1/9) [x - (-2)]

[y + 2] = (-1/9) [x + 2]

9 (y + 2) = -1 (x + 2)

9 y + 18 = - x - 2

x + 9 y + 18 + 2 = 0

x + 9 y + 20 = 0

tangent and normal question6 tangent and normal question6

- Back to worksheet
- Rolle's theorem
- Lagrange's theorem
- Finding increasing or decreasing interval
- Increasing function
- Decreasing function
- Monotonic function
- Maximum and minimum
- Examples of maximum and minimum