# Given that x and y vary so that ax + by= c , where a,b,c are constant. Show that the minimum value of x^2 + y^2 is c^2/ a^2 + b^2 ?

Feb 22, 2017

The minimum value is ${c}^{2} / \left({a}^{2} + {b}^{2}\right)$

#### Explanation:

The minimum of ${x}^{2} + {y}^{2}$ restricted to $a x + b y = c$ is obtained by minimizing

$f \left(x\right) = {x}^{2} + {\left(\frac{c - a x}{b}\right)}^{2}$ The condition of relative minimum is

$\frac{\mathrm{df}}{\mathrm{dx}} = 0$ at $x = {x}_{0}$ and

$\frac{{d}^{2} f}{{\mathrm{dx}}^{2}} > 0$ at $x = {x}_{0}$

Then

$\frac{\mathrm{df}}{\mathrm{dx}} = 2 x - 2 \left(\frac{a}{b}\right) \left(\frac{c - a x}{b}\right) = 0 \to {x}_{0} = \frac{a c}{{a}^{2} + {b}^{2}}$

substituting this value into $f \left(x\right)$ we have

$f \left({x}_{0}\right) = {c}^{2} / \left({a}^{2} + {b}^{2}\right)$

Also

$\frac{{d}^{2} f}{{\mathrm{dx}}^{2}} = 2 + \frac{2 {a}^{2}}{b} ^ 2 > 0$

Feb 23, 2017

${\left({x}^{2} + {y}^{2}\right)}_{\min} = {c}^{2} / \left({a}^{2} + {b}^{2}\right) .$

#### Explanation:

Let $O \left(0 , 0\right)$ be the Origin, and, $\left(x , y\right)$ be the General

Point in the Plane ${\mathbb{R}}^{2.}$

If L={(x,y) : ax+by=c; a,b,c in RR}, then, we know that, $L$ is

a Family of Lines in the plane, where, ${a}^{2} + {b}^{2} \ne 0.$

Let us note that, $c = 0 \iff O \left(0 , 0\right) \in L .$

Now, the Distance $O P = \sqrt{{x}^{2} + {y}^{2}} \Rightarrow {x}^{2} + {y}^{2} = O {P}^{2.}$

Accordingly, when $\left({x}^{2} + {y}^{2}\right)$ is Minimum, so is the Distance

$O P .$

Thus, in this new scenario, our Problem is to find the

Minimum Distance from the Origin $O \left(0 , 0\right) \text{ to the Line } L .$

From Geometry, we know that the minimum distance of a point

$\left(x ' , y '\right)$ to a line : $A x + b y + C = 0$ is the $\bot \text{-distance}$ between

them, given by, $| A x ' + b y ' + C \frac{|}{\sqrt{{A}^{2} + {b}^{2}}}$.

Accordingly, ${\left\{\sqrt{{x}^{2} + {y}^{2}}\right\}}_{\min} = | c \frac{|}{\sqrt{{a}^{2} + {b}^{2}}} , \mathmr{and} ,$

${\left({x}^{2} + {y}^{2}\right)}_{\min} = {c}^{2} / \left({a}^{2} + {b}^{2}\right) \ldots \ldots \ldots \ldots \ldots \ldots \ldots . \left(\star\right)$

We conclude with a note that, in case, $O \left(0 , 0\right) \in L , i . e . , c = 0 ,$

${\left({x}^{2} + {y}^{2}\right)}_{\min} = 0 ,$ Geometrically, and, in accordance with

$\left(\star\right)$ as well.

Enjoy Maths.!