The chord of the curve y = x2 + 2ax + b, joining the points where x = α and x = β, is parallel to the tangent to the curve at abscissa x =

Question

The chord of the curve y = x2 + 2ax + b, joining the points where x = α and x = β, is parallel to the tangent to the curve at abscissa x = (A) (a+b/2) (B) (2a+b/3) (C) (2α+β/3) (D) (α+β/2)

Tardigrade Admin · Accepted Answer

Given curve, y=x2+2 a x+b At x=α, y =α2+2 a α+b and at x =β, y=β2+2 a β + b ∴ Slope of line joining (α, α2+2 a α+b) and (β, β2+2 a β+b) is =((α2+2 a α+b)-(β2+2 a β+b)/α-β) =(α2+2 a α+b-β2-2 a β-b/α-β) =((α2-β2)+2 a(α-β)/α-β) =((α-β)(α+β)+2 a(α-β)/α-β) =α+β+2 a Slope of given curve =(d y/d x) =2 x+2 a Now, according to question, tangent is parallel to the chord. Therefore, 2 x+2 a =α+β+2 a ⇒ 2 x =α+β ⇒ x=(α+β/2)

Q. The chord of the curve $y = x^{2} + 2 a x + b$ , joining the points where $x = α$ and $x = β$ , is parallel to the tangent to the curve at abscissa $x =$

$\frac{a + b}{2}$

$\frac{2 a + b}{3}$

$\frac{2 α + β}{3}$

$\frac{α + β}{2}$

Q. The chord of the curve y=x2+2ax+b, joining the points where x=α and x=β, is parallel to the tangent to the curve at abscissa x=

2a+b​

32a+b​

32α+β​

2α+β​

Q. The chord of the curve $y = x^{2} + 2 a x + b$ , joining the points where $x = α$ and $x = β$ , is parallel to the tangent to the curve at abscissa $x =$

$\frac{a + b}{2}$

$\frac{2 a + b}{3}$

$\frac{2 α + β}{3}$

$\frac{α + β}{2}$