Question

The triangle formed by the tangent to the parabola $y=x^2$ at the point whose abscissa is k, where $[1,2]$ the y-axis and the straight line $y=k^2$ has greatest area if k is equal to

• A. $1$
• B. $3$
• C. $2$
• D. none of these

Answer 1

The correct option is C $2$

The triangle formed by tangent to parabola $y=x^2$ whose abscissa is $k$ where $k\in [1,2]$,the y axis and the straight line $y=k^2$ has greatest area if $k$ is equal to:

$\frac{y}{2}+\frac{k^2}{2}=2kx2kx-y=k^2..........\left(1\right)x=\mathrm{0..............}\left(2\right)y=k^2.............\left(3\right)A\equiv (0,-k^2)B\equiv (k,k^2)C\equiv (0,k^2)$

Area $=\frac{1}{2}\left[x_1\right(y_2-y_3)+x_2(y_3-y_1)+x_3(y_1-y_2\left)\right]$

$=\frac{1}{2}\left[2k^3\right]=k^{3}k\in [1,2]$

Area is maximum when $k$ is max

$k=2$