Question

If the line y = 4x – 5 touches to the curve y² = ax 3+ bat the point (2, 3) then 7a + 2b = ______.

Options

• 0

• 1

• –1

• 2

Answer 1

If the line y = 4x – 5 touches to the curve y² = ax 3+ bat the point (2, 3) then 7a + 2b = 0.

Explanation:

Line y = 4x – 5 `rightarrow` slope of line m = 4 ...(i)

curve y² = ax³ + b

∴ differentiating w.r.t. ‘x’

`2y dy/dx` = 3ax²

`dy/dx = (3ax^2)/(2y)` = slope of tangent

∴ `dy/dx|_((2"," 3)) = (3a xx 4)/(2 xx 3)` = 2a  ...(ii)

∴ from (i) and (ii), we get

4 = 2a `implies` a = 2

Since, (2, 3) is a point on the curve : y² = ax³ + b.

∴ (3)² = 2(2)³ + b

`\implies` b = – 7

∴ 7a + 2b = 7 × 2 + 2(–7) = 0