If the image of the point (–4,5) in the line x + 2y = 2 lies on the circle , then r is equal to : [2024]

Question

If the image of the point (–4,5) in the line x + 2y = 2 lies on the circle ${(x + 4)}^{2} + {(y - 3)}^{2} = r^{2}$ , then r is equal to : [2024]

Smart Achievers · Accepted Answer

(4)

$\frac{x + 4}{1} = \frac{y - 5}{2} = \frac{- 2 (- 4 + 2 \times 5 - 2)}{1 + 4} = \frac{- 8}{5}$

$\Rightarrow x = - 4 - \frac{8}{5} = - \frac{28}{5}, y = 5 - \frac{16}{5} = \frac{9}{5}$

$∴ Image is (\frac{- 28}{5}, \frac{9}{5})$

Image lies on circle ${(x + 4)}^{2} + {(y - 3)}^{2} = r^{2}$

${(\frac{- 28}{5} + 4)}^{2} + {(\frac{9}{5} - 3)}^{2} = r^{2}$

$\Rightarrow \frac{64}{25} + \frac{36}{25} = r^{2} \Rightarrow r^{2} = 4 \Rightarrow r = 2$

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