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 curve at P is :

(1) x + 3y – 62 = 0 

(2) x – 3y – 11 = 0 

(3) x – 3y + 22 = 0 

(4) x + 3y + 26 = 0

Answer 1

Answer (4) x + 3y + 26 = 0

Let L be the common normal to parabola y = x² + 7x + 2 and line y = 3x – 3

⇒ slope of tangent of y = x² + 7x + 2 at P = 3

⇒ 2x + 7 = 3 ⇒ x = –2 ⇒ y = –8

So P(–2, –8)

Normal at P : x + 3y + C = 0

⇒ C = 26 (P satisfies the line)