Question

Let A (-2, 0) and B(1,3) be two fixed points. Find coordinates of a variable point P on parabola $y^2=4x$ such that area of $\Delta PAB$ is minimum

• A. $(0,0)$
• B. $(4,4)$
• C. $(1,2)$
• D. None of these

Answer 1

The correct option is C $(1,2)$

Let $A(-2,0)$ and $B(1,3)$ be two fixed points find coordinates of a variable point P on parabola $y^2=4x$ such area of $△PAB$ is minimum.

Let $P\equiv (t^2,2t)$

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

$=\frac{1}{2}|-2(3-2t)+1(2t-0)+t^2(0-3\left)\right|=\frac{3}{2}|t^2-2t+2|$

Let $A=t^2-2t+2$ is minimum

$\frac{dA}{dt}=2t-2=0t=1P\equiv (1,2)$