If 12a + 5b = 9,where a, b in mathbb{R},then | vedclass

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(A) The expression $a^2 + b^2$ represents the square of the distance from the origin $(0, 0)$ to a point $(a, b)$ on the line $12a + 5b = 9$.The minim

Vedclass · Accepted Answer

$\frac{81}{169}$

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(A) The expression $a^2 + b^2$ represents the square of the distance from the origin $(0, 0)$ to a point $(a, b)$ on the line $12a + 5b = 9$.The minimum value of $a^2 + b^2$ is the square of the perpendicular distance from the origin to the line $12a + 5b - 9 = 0$.The formula for the perpendicular distance $d$ from $(x_1, y_1)$ to $Ax + By + C = 0$ is $d = \frac{|Ax_1 + By_1 + C|}{\sqrt{A^2 + B^2}}$.Here,$A = 12$,$B = 5$,$C = -9$,$x_1 = 0$,and $y_1 = 0$.$d = \frac{|12(0) + 5(0) - 9|}{\sqrt{12^2 + 5^2}} = \frac{|-9|}{\sqrt{144 + 25}} = \frac{9}{\sqrt{169}} = \frac{9}{13}$.The minimum value of $a^2 + b^2$ is $d^2 = \left(\frac{9}{13}\right)^2 = \frac{81}{169}$.

If $12a + 5b = 9$,where $a, b \in \mathbb{R}$,then the minimum value of $a^2 + b^2$ is:

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