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Global Maxima

For the point...

Question

For the points on the circle $x^{2} + y^{2} - 2 x - 2 y + 1 = 0$ , the sum of the maximum and minimum values of $4 x + 3 y$ is

\frac{26}{3}

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10

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12

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14

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Solution

The correct option is D $14$
The center and radius of the circle $x^{2} + y^{2} - 2 x - 2 y + 1 = 0$ are (1, 1) and 1 respectively
$∴$ any point on it can be written as $(1 + cos θ, 1 + sin θ)$
$4 x + 3 y = 4 (1 + cos θ) + 3 (1 + sin θ)$
$4 x + 3 y = 7 + 4 c o s θ + 3 s i n θ$
We have,
$- 5 \leq 4 c o s θ + 3 s i n θ \leq 5$
Hence, minimum value $= 7 - 5 = 2$ and maximum value $= 7 + 5 = 12$
Hence, the required sum $= 14$

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For the points on the circle x2+y2−2x−2y+1=0, the sum of the maximum and minimum values of 4x+3y is

For the points on the circle $x^{2} + y^{2} - 2 x - 2 y + 1 = 0$ , the sum of the maximum and minimum values of $4 x + 3 y$ is