Question

Let $f(x,y)=\sqrt{x^2+y^2}+\sqrt{x^2+y^2-2x+1}+\sqrt{x^2+y^2-2y+1}+\sqrt{x^2+y^2-6x-8y+25}\forall x,yϵR,then$

• A. Minimum value of f(x, y) = 5 + $\sqrt{2}$
• B. Minimum value of f(x, y) = 5 - $\sqrt{2}$
• C. Minimum value occurs of f(x, y) for x =$\frac{3}{7}$
• D. Minimum value occurs of f(x, y) for y =$\frac{4}{7}$

Answer 1

Answer: (A), (C), (D)

The correct options are
A

Minimum value of f(x, y) = 5 + $\sqrt{2}$

C

Minimum value occurs of f(x, y) for x =$\frac{3}{7}$

D

Minimum value occurs of f(x, y) for y =$\frac{4}{7}$

Let O(0, 0), A = (1, 0), B = (3, 4), C = (0, 1)
Now for (x, y) to be minimum, (x, y) must be the point of intersection of OB and AC

$\therefore f(x,y{)}_{min}=OB+AC=5+\sqrt{2}$
which occurs at
$x=\frac{3}{7},y=\frac{4}{7}$