Question

If x + 4y = 14 is a normal to the curve 2y³ = αx³ - β at (2, 3), then the value of α + β is

(a) 9

(b) –5

(c) 7

(d) –7

Answer 1

Correct option is (a) 9

Given equation of curve is 

y² = αx³ − β

Since, it passes through (2, 3)

4 = 8α − β

\(\frac{dy}{dx} = \frac{3\alpha x^2}{2y}\)

Slope of tangent at (2, 3) is 2α

Slope of normal  at (2, 3) = \(- \frac 1{2\alpha}\)

Given equation of normal is x + 4y = 14

Slope of normal is \(- \frac 14\)

⇒ α = 2

⇒ β = 7

Hence α + β = 9