If h1,h2,h3 are lengths of altitudes of ΔleABC and r is the in radius then h1+rh1-r+h2+rh2−r+h3+rh3−r is

h1=2Δa,h2=2Δb,h3=2Δch1+rh1-r=2Δa+Δs2Δa-Δs=2s+a2s-ah1+rh1−r+h2+rh2−r+h3+rh3−r=∑2s+a2s−a=∑4s2s−a−3.........(1)A.M≥H.MBut 4s2s−a+4s2s−b+4s2s−c3≥32s−a4s+2s−b4s+2s−c4s=3∑4s2s−a ≥9From (1)⇒h1+rh1−r+h2+rh2−r+h3+rh3−2=∑4s2s−a−3≥9−3=6

If h1,h2,h3 are lengths of altitudes of ΔleABC and r is the in radius then h1+rh1-r+h2+rh2−r+h3+rh3−r is

h1=2Δa,h2=2Δb,h3=2Δch1+rh1-r=2Δa+Δs2Δa-Δs=2s+a2s-ah1+rh1−r+h2+rh2−r+h3+rh3−r=∑2s+a2s−a=∑4s2s−a−3.........(1)A.M≥H.MBut 4s2s−a+4s2s−b+4s2s−c3≥32s−a4s+2s−b4s+2s−c4s=3∑4s2s−a ≥9From (1)⇒h1+rh1−r+h2+rh2−r+h3+rh3−2=∑4s2s−a−3≥9−3=6

AI Mentor

Check Your IQ

Free Expert Demo

Try Test

Courses

Dropper NEET Course Dropper JEE Course Class - 12 NEET Course Class - 12 JEE Course Class - 11 NEET Course Class - 11 JEE Course Class - 10 Foundation NEET Course Class - 10 Foundation JEE Course Class - 10 CBSE Course Class - 9 Foundation NEET Course Class - 9 Foundation JEE Course Class -9 CBSE Course Class - 8 CBSE Course Class - 7 CBSE Course Class - 6 CBSE Course

Offline Centres

If $h_{1}, h_{2,} h_{3}$ are lengths of altitudes of $Δ^{le} ABC$ and r is the in radius then $\frac{h_{1} +r}{h_{1} -r} + \frac{h_{2} + r}{h_{2} - r} + \frac{h_{3} + r}{h_{3} - r}$ is

see full answer

Your Exam Success, Personally Taken Care Of

1:1 expert mentors customize learning to your strength and weaknesses – so you score higher in school , IIT JEE and NEET entrance exams.

An Intiative by Sri Chaitanya

$> 6$

$\geq 6$

$< 6$

$\leq 6$

answer is A.

(Unlock A.I Detailed Solution for FREE)

Best Courses for You

JEE

NEET

Foundation JEE

Foundation NEET

CBSE

Detailed Solution

$h_{1} = \frac{2 Δ}{a}, h_{2} = \frac{2 Δ}{b}, h_{3} = \frac{2 Δ}{c}$
$\frac{h_{1} + r}{h_{1} - r} = \frac{\frac{2 Δ}{a} + \frac{Δ}{s}}{\frac{2 Δ}{a} - \frac{Δ}{s}} = \frac{2 s + a}{2 s - a}$
$\frac{h_{1} + r}{h_{1} - r} + \frac{h_{2} + r}{h_{2} - r} + \frac{h_{3} + r}{h_{3} - r} = \sum \frac{2 s + a}{2 s - a} = \sum \frac{4 s}{2 s - a} - 3 ......... (1)$
$A . M \geq H . M$
But $\frac{\frac{4 s}{2 s - a} + \frac{4 s}{2 s - b} + \frac{4 s}{2 s - c}}{3} \geq \frac{3}{\frac{2 s - a}{4 s} + \frac{2 s - b}{4 s} + \frac{2 s - c}{4 s}} = 3$
$\sum \frac{4 s}{2 s - a} \geq 9$
From (1) $\Rightarrow \frac{h_{1} + r}{h_{1} - r} + \frac{h_{2} + r}{h_{2} - r} + \frac{h_{3} + r}{h_{3} - 2} = \sum \frac{4 s}{2 s - a} - 3$ $\geq 9 - 3 = 6$