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Question

The line y=x is a tangent to the parabola y=ax2+bx+c at the point x=1.If the parabola passes through the point (−1,0), then determine a,b,c.

A
a=12,b=14,c=13.
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B
a=14,b=12,c=14.
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C
a=2,b=1,c=4.
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D
a=4,b=2,c=4.
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Solution

The correct option is B a=14,b=12,c=14.
Given equation of parabola is
y=ax2+bx+c
dydx=2ax+b
Slope of tangent to the curve at x=1 is 2a+b
Given tangent is y=x . Slope of this tangent is 1.
So, 2a+b=1 ...(1)
Since, the parabola passes through (1,0)
ab+c==0 ...(2)
Given y=x is a tangent at x=1
y=1.
Hence (1,1) lies both on tangent and parabola
a+b+c=1 ...(3)
Solving (1), (2) and (3), we get
a=14,b=12,c=14.

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