The chord of the curve y - x2 - 2ax - b- joining the points where x - - and x - - is parallel to the tangent to the curve at abscissa x -

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Equation of Tangent at a Point (x,y) in Terms of f'(x)

The chord of ...

Question

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}

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\frac{2 a + b}{3}

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\frac{2 α + β}{3}

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\frac{α + β}{2}

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Solution

The correct option is D $\frac{α + β}{2}$
Given curve is $y = x^{2} + 2 a x + b$
Differentiate above equation w.r.t. $x$ we get
$\frac{d y}{d x} = 2 x + 2 a$ is the equation of tangent to the curve
But tangent to the curve is parallel to the chord of the curve which joins the points $x = α$ and $x = β$
$∴$ Tangent to this curve is $= (α + β) + 2 a$
$∴ 2 x + 2 a = (β + α) + 2 a$
$⟹ x = \frac{α + β}{2}$

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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=

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 $x =$