Finding maximum value of an expression $3a+4b$ where $(a,b)$ lies on unit circle.

Question

If the point $(a,b)$ lies on circle $x^2+y^2=1$ then maximum value of $3a+4b$ is ?

Answer 1

Put $x=sin (t),y=cos (t) $ thus we need max value of $3sin (t)+4cos (t)\leq \sqrt {3^2+4^2}=5$

Answer 2

$$3a+4b=(3,4).(a,b)\le \sqrt{25\times (a^2+b^2)}=5$$

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In general $$(x.y)^2 \le |x|.|y|$$

and equality holds iff $x,y$ be parallel vectors.