Little bit Logical

Algebra Level pending

a and b are integers in the eqation and underroot m is the smallest value of the eqation

a ( a − 18 ) + b ( b − 32 ) + 337 + a ( a − 14 ) + b ( b − 10 ) + 74 = m \sqrt{a(a-18)+b(b-32)+337}+ \sqrt{a(a-14)+b(b-10)+74}=\sqrt{m}

Find the value of m


The answer is 125.

Avinash Singh by Avinash Singh , 21, India

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1 solution

Avinash Singh
Mar 20, 2015

a ( a − 18 ) + b ( b − 32 ) + 337 + a ( a − 14 ) + b ( b − 10 ) + 74 = m \sqrt{a(a-18)+b(b-32)+337}+\sqrt{a(a-14)+b(b-10)+74}=\sqrt{m} a 2 − 18 a + b 2 − 32 b + 337 + a 2 − 14 a + b 2 − 10 b + 74 = m \sqrt{a^2-18a+b^2-32b+337}+\sqrt{a^2-14a+b^2-10b+74}=\sqrt{m} ] ( a − 9 ) 2 + ( b − 16 ) 2 + ( a − 7 ) 2 + ( b − 5 ) 2 = m \sqrt{(a-9)^2+(b-16)^2}+\sqrt{(a-7)^2+(b-5)^2}=\sqrt{m}

Now see the pattern here...... ABC is triangle with the coordinate of A,B,C 

                             A(9,16)

                   B (7,5)           C(a,b)

the eqation take form as

A C + B C = m AC+BC=\sqrt{m} (m is the smallest value so the three point are collinear)

A C + B C = ( 9 − 7 ) 2 + ( 16 − 5 ) 2 AC+BC=\sqrt{(9-7)^2+(16-5)^2}

A C + B C = 125 AC+BC=\sqrt{125}

so m=125

3 Helpful 0 Interesting 0 Brilliant 0 Confused

Good one.....keep it up 😃

Sakanksha Deo - 6 years, 2 months ago

Very well executed and thought of Avinash ! Although i did it by observing and putting the values.Lowest value did come for a as 9 and b as 16.

rahul saxena - 6 years, 2 months ago

That problem deserved a like

James Wilson - 3 years, 8 months ago

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