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Question

# The chord joining the points where x = p and x = q on the curve y=ax2+bx+c is parallel to the tangent at the point on the curve whose abscissa is

A
p+q2
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B
pq2
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C
pq2
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D
pq
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Solution

## The correct option is A p+q2Given f(x)=ax2+bx+c is continuous on [p,q] andf′(x)=2ax+b is differentiable on (p,q).Therefore, Lagrange's mean value theorem can be applied.Lagrange's mean value theorem states that if f(x) be continuous on [a,b] and differentiable on (a,b) then there exists some c between a and b such that f′(c)=f(b)−f(a)b−aTherefore, f′(c)=(aq2+bq+c)−(ap2+bp+c)q−p⟹2ac+b=a(q2−p2)+b(q−p)q−p⟹2ac+b=a(q+p)+b⟹2ac=a(p+q)⟹2c=p+q⟹c=p+q2Therefore, the abscissa of the point on the curve where the tangent is parallel to the chord is p+q2

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