Question

Let P(h, k) be a point on the curve y = x² + 7x + 2, nearest to the line, y = 3x – 3. Then the equation of the normal to the curve at P is ______.

Options

• x + 3y – 62 = 0

• x – 3y – 11 = 0

• x – 3y + 22 = 0

• x + 3y + 26 = 0

Answer 1

Let P(h, k) be a point on the curve y = x² + 7x + 2, nearest to the line, y = 3x – 3. Then the equation of the normal to the curve at P is x + 3y + 26 = 0.

Explanation:

P(h, k) y = x² + 7x + 2

k = h² + 7h + 2  ...(i)

Point and line distance

D = `(3h - k - 3)/sqrt(10)`

From (i),

D = `(3h - h^2 - 7h - 2 - 3)/sqrt(10)`

`(dD)/(dh) = (3 - 2h - 7)/sqrt(10)`

For maxima and minima

`(dD)/(dh)` = 0

h = –2

From (i),

k = (–2)² – 14 + 2 = –8

`(d^2D)/(dh^2)` = –2    Maxima

Normal at (–2, –8)

y – y₁ = `- 1/(dy/dx)_((-2"," -8)) (x - x_1)`

y + 8 = `1/((2x + 7)_((-2"," - 8))) (x + 2)`

y + 8 = `(-1)/3(x + 2)`

3y + 24 = –x – 2

x + 3y + 26 = 0