Question

The image of the point $(3,5)$ in the line$x–y+1=0$, lies on :

• A. ${(x–2)}^2+{(y–4)}^2=4$
• B. ${(x–4)}^2+{(y+2)}^2=16$
• C. ${(x–4)}^2+{(y–4)}^2=8$
• D. ${(x–2)}^2+{(y–2)}^2=12$

Answer 1

The correct option is A

${(x–2)}^2+{(y–4)}^2=4$

Explanation for correct option

Step 1.Find the image of the point

We know that if $B\left(x,y\right)$ is the image of a point $P\left(p,q\right)$ with respect to a line $ax+by+c=0$ then

$\frac{\mathbf{x}\mathbf{-}\mathbf{p}}{\mathbf{a}}\mathbf{=}\frac{\mathbf{y}\mathbf{-}\mathbf{q}}{\mathbf{b}}\mathbf{=}\frac{\mathbf{-}\mathbf{2}\left(\mathrm{ap}+\mathrm{bq}+c\right)}{{\mathbf{a}}^{\mathbf{2}}\mathbf{+}{\mathbf{b}}^{\mathbf{2}}}$

Image of the point$(3,5)$ in the line$x–y+1=0$, is

$$\begin{gathered} \frac{x-3}{1}=\frac{y-5}{-1}=\frac{-2\left(3-5+1\right)}{2}=1 \\ \therefore x=4,y=4 \end{gathered}$$

Image of the point $(3,5)$ is $\left(4,4\right)$

For option A

Image of the point $(3,5)$ is $\left(4,4\right)$

If the image lies on the circle ${(x–2)}^2+{(y–4)}^2=4$, then the image must satisfy the equation of the circle

Since,

$$\begin{gathered} {\left(4-2\right)}^2+{\left(4-4\right)}^2 \\ =2^2+0 \\ =4 \end{gathered}$$

So Option A is correct

Step 3. Explanation for incorrect option

Image of the point $(3,5)$ is $\left(4,4\right)$

For option B

If the image lies on the circle ${(x–4)}^2+{(y+2)}^2=16$, then the image must satisfy the equation of the circle

$$\begin{gathered} {(x–4)}^2+{(y+2)}^2 \\ ={\left(4-4\right)}^2+{\left(4+2\right)}^2 \\ =36\ne 16 \end{gathered}$$

So Option B is incorrect

For option C

If the image lies on the circle ${(x–4)}^2+{(y–4)}^2=8$, then the image must satisfy the equation of the circle

$$\begin{gathered} {(x–4)}^2+{(y–4)}^2 \\ ={(4–4)}^2+{(4–4)}^2 \\ =0\ne 8 \end{gathered}$$

So option C is incorrect

For option D

If the image lies on the circle ${(x–2)}^2+{(y–2)}^2=12$, then the image must satisfy the equation of the circle

$$\begin{gathered} {(x–2)}^2+{(y–2)}^2 \\ ={(4–2)}^2+{(4–2)}^2 \\ =8\ne 12 \end{gathered}$$

So option D is incorrect

Hence, option A is correct.