Given that \(\displaystyle x^2\,+\,y^2\,=\,14x\,+\,6y\,+\,6\), what is the largest possible value that \(\displaystyle 3x\,+\,4y\) can have?

Obviously, Google will give us this and YouTube will give us this to determine the correct answer of 73.

My solution uses this. If 3x + 4y = C then we need to determine the maximum value for C. 0 = xÂ² + yÂ² - 14x - 6y - 6 = (x - 7)Â² - 7Â² + (y - 3)Â² - 3Â² - 6 i.e. (x - 7)Â² + (y - 3)Â² = 8Â².

Let b = C/4 or C = 4b where 0 = 3x/4 + y - b.

\(\displaystyle \frac{|\frac{3\,\times\,7}{4}\,+\,3\,-\,b|}{\sqrt{\frac{9}{16}\,+\,1}}\,=\,8\) i.e. the distance from center (7, 3) to the point of tangency equals to the radius 8.

\(\displaystyle |\frac{33}{4}\,-\,b|\,=\,10\) i.e. \(\displaystyle b\,-\,\frac{33}{4}\,=\,10\) so b = 10 + 33/4 = 40/4 + 33/4 = (40 + 33)/4 = 73/4 â‰ -7/4 (âˆµ b = 73/4 > -7/4).

âˆ´ C = 4 Ã— 73/4 = 73 â‰ -7 (âˆµ C = 73 > -7).

Obviously, Google will give us this and YouTube will give us this to determine the correct answer of 73.

My solution uses this. If 3x + 4y = C then we need to determine the maximum value for C. 0 = xÂ² + yÂ² - 14x - 6y - 6 = (x - 7)Â² - 7Â² + (y - 3)Â² - 3Â² - 6 i.e. (x - 7)Â² + (y - 3)Â² = 8Â².

Let b = C/4 or C = 4b where 0 = 3x/4 + y - b.

\(\displaystyle \frac{|\frac{3\,\times\,7}{4}\,+\,3\,-\,b|}{\sqrt{\frac{9}{16}\,+\,1}}\,=\,8\) i.e. the distance from center (7, 3) to the point of tangency equals to the radius 8.

\(\displaystyle |\frac{33}{4}\,-\,b|\,=\,10\) i.e. \(\displaystyle b\,-\,\frac{33}{4}\,=\,10\) so b = 10 + 33/4 = 40/4 + 33/4 = (40 + 33)/4 = 73/4 â‰ -7/4 (âˆµ b = 73/4 > -7/4).

âˆ´ C = 4 Ã— 73/4 = 73 â‰ -7 (âˆµ C = 73 > -7).

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