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  4. If a,b∈ R satisfy the equation a2+4b2-4=0, then the minimum value of (2 a + 3 b) will be

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Q. If $a,b\in R$ satisfy the equation $a^{2}+4b^{2}-4=0,$ then the minimum value of $\left(2 a + 3 b\right)$ will be

NTA AbhyasNTA Abhyas 2022

Solution:

Given equation is $\frac{a^{2}}{4}+b^{2}=1$
Putting $a=2\cos \theta ,b=\sin⁡\theta $ , we get,
$2a+3b$ $=2\left(2 \cos \theta \right)+3\left(\sin ⁡ \theta \right)=4\cos\theta +3\sin\theta $
$\Rightarrow $ Minimum value $=-\sqrt{4^{2} + 3^{2}}=-5$