Oxford University MAT 2020 Solved Papers

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Question : 5 of 10

Marks: +1, -0

A line is tangent to the parabola

y=x^2

at the point

(a, a^2)

where

a>0

. The area of the region bounded by the parabola, the tangent line, and the

x

-axis equals

$\;{a^2}/{3}$ ,
$\;{2 a^2}/{3}$ ,
$\;{a^3}/{12}$ ,
$\;{5 a^3}/{6}$ ,
$\;{a^4}/{10}$ .

Solution:

The tangent is

y=2 a x-a^2

, which crosses the

x

-axis at

(a / 2,0)

.

The area is

∫↙0^a x^2 {d} x-(

area of APQ) where

A

is

(a, a^2), P

is

(a, 0)

and

Q

is

(a \/ 2,0)

. This is

\;{a^3}/{3}-\;{a^3}/{4}=\;{a^3}/{12}

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