Find the image of the point (2, 1) with respect to the line mirro… - Vidyadip

Question

Find the image of the point (2, 1) with respect to the line mirror x + y - 5 = 0.

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Answer

Let the image of the point P(2, 1) in the mirror AB be $\text{Q}(\alpha,\beta).$ Then, PQ is perpendicular bisected at R. The coordinates of R are $\Big(\frac{\alpha+2}{2},\frac{\beta+1}{2}\Big)$ And lie on the line x + y - 5 = 0 $\Big(\frac{\alpha+2}{2}\Big)+\Big(\frac{\beta+1}{2}\Big)-5=0$ $\alpha+2+\beta+1-10=0$ $\alpha+\beta=7 \ ...(1)$ Since PQ is $\perp$ to AB (Slope of AB) × (Slope of PQ) = -1 $-1\times\Big(\frac{\beta-1}{\alpha-2}\Big)=-1$ $\beta-1=\alpha-2$ $\beta-\alpha=-1 \ ...(2)$ Solving (1) and (2), we get $\alpha=5$ and $\beta=2$ $\therefore$ Image of (1, 2) in x + y - 5 = 0 is (4, 3).

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