A point on the hypotenuse of a right angled triangle is at distances 'a' and 'b' from the sides - Maths - Application of Derivatives

Question

A point on the hypotenuse of a right angled triangle is at
distances 'a' and 'b' from the sides
Show that the length of the hypotenuse is at least
{ a raised to 2/3 +
b raised to 2/3}the whole raised to
3/2

Harleen Birdi · Accepted Answer

Let AOB be a right triangle with hypotenuse AB such that a point P on AB is distance a and b from OA and OB respectively. i.e. PL = a and PM = b

Let ∠OAB = θ Then,

AP = a cosec θ and BP = b sec θ

Let l be the length of the hypotenuse AB. Then

l = AP + BP

⇒ l = a cosec θ + b sec θ

$\Rightarrow \frac{d l}{d θ} = - a \cos e c θ \cot θ + b \sec θ tan θ and \frac{d^{2} l}{d θ^{2}} = a \cos e c^{3} θ + a \cos e c {θ cot}^{2} θ + b \sec^{2} θ + b \sec {θ tan}^{2} θ$

For maximum or minimum, we must have

$\frac{d l}{d θ} = 0 \Rightarrow - a \cos e c θ cot θ + b \sec θ \tan θ = 0 \Rightarrow - \frac{a \cos θ}{\sin^{2} θ} + \frac{b \sin θ}{\cos^{2} θ} = 0 \Rightarrow \tan^{3} θ = \frac{a}{b} \Rightarrow \tan θ = (\frac{a}{b})^{\frac{1}{3}} \Rightarrow \sin θ = \frac{a^{\frac{1}{3}}}{\sqrt{a^{\frac{2}{3}} + b^{\frac{2}{3}}}} and cos θ = \frac{b^{\frac{1}{3}}}{\sqrt{a^{\frac{2}{3}} + b^{\frac{2}{3}}}}$

Clearly, $\frac{d^{2} l}{d θ} > 0 for tan θ = (\frac{a}{b})^{\frac{1}{3}}$

Thus, l is minimum when $\tan θ = (\frac{a}{b})^{\frac{1}{3}}$

The minimum value of l is given by

$l = a \cos e c θ + b \sec θ = a \sqrt{1 + \cot^{2} θ} + b \sqrt{1 + \tan^{2} θ} \Rightarrow l = a \sqrt{1 + (\frac{b}{a})^{\frac{2}{3}}} + b \sqrt{1 + (\frac{a}{b})^{\frac{2}{3}}} \Rightarrow l = (a^{\frac{2}{3}} + b^{\frac{2}{3}})^{\frac{3}{2}}$

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A point on the hypotenuse of a right angled triangle is at distances 'a' and 'b' from the sides . Show that the length of the hypotenuse is at least { a raised to 2/3 + b raised to 2/3}the whole raised to 3/2

A point on the hypotenuse of a right angled triangle is at distances 'a' and 'b' from the sides .

Show that the length of the hypotenuse is at least

{ a raised to 2/3 + b raised to 2/3}the whole raised to 3/2