If real a and b satisfy 17(a^2+b^2)-30ab-16=0 then find the

bacasauvfl

bacasauvfl

Answered question

2022-04-24

If real a and b satisfy 17(a2+b2)30ab16=0 then find the maximum value of 16a2+4b216ab12a+6b+9

Answer & Explanation

Jonas Dickerson

Jonas Dickerson

Beginner2022-04-25Added 22 answers

Step 1
16a2+4b2-16ab-12a+6b+9=(4a-2b)2-2(4a-2b)·(3/2)+9

=(4a-2b-3/2)2+9+(3/2)2

Let 4a2b=2cb=2ac
From the given condition
0=17a2+17(2ac)230a(2ac)16
=a2(17+6860)+a(60c68c)+17c216=0
which is a quadratic equation in a
As a is real, the discriminant must be 0
Use this fact to find the range of values of c

Klanglinkmgk

Klanglinkmgk

Beginner2022-04-26Added 13 answers

Step 1
The hint.
Let 2ab=k
Thus, we need to find a maximal value of 4k26k+9.
Now, find all values of k, for which the equation
k2(17(a2+b2)30ab)=16(2ab)2
has solutions and choose from these values such that 4k26k+9 will get a maximal value.
I got that it happens for k=52

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