ISI2019-MMA-17

Question

The reflection of the point $(1,2)$ with respect to the line $x + 2y =15$ is

• A. $(3,6)$
• B. $(6,3)$
• C. $(5,10)$
• D. $(10,5)$

Comments

how (5,10)?

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From the few integral solutions of the equation, one of them is (3,6). Now reflection of (1,2) wrt (3,6) would be (5,10). This is how I solved in the exam, might not be the proper method though

Answer 1

The formula for finding the foot of the perpendicular from a point $(x_1,y_1)$ to the line $ax+by+c=0$ is given by:

$$\displaystyle{\frac{x − x_1}{a} = \frac{y − y_1}{b} = \frac{−(ax_1 + by_1 + c)}{a^2 + b^2}}$$

Note: I didn't include the proof for above formula as the answer would have become too long. If someone wants the proof for above formula, refer this article https://math.stackexchange.com/questions/1013230/how-to-find-coordinates-of-reflected-point.

For finding the image of the point in the same line, we just multiply the rightmost term by $2$. The image of the point is at the same distance from the line as the point itself is from the line. So, we have to multiply it by $2$.

So, the image of the point $(x_1, y_1)$ in the line $ax_1 + by_1 + c = 0$ is given by:

$$\displaystyle{\frac{x − x_1}{a} = \frac{y − y_1}{b} = \frac{−2(ax_1 + by_1 + c)}{a^2 + b^2}}$$

Applying the above formula, we get $$\displaystyle{\frac{x − 1}{1} = \frac{y − 2}{2} = \frac{−2(1 + 2*2 - 15)}{1^2 + 2^2}} = \frac{20}{5} = 4$$

$$\displaystyle{\frac{x − 1}{1} = 4} \implies x = 5$$

$$\displaystyle{\frac{y − 2}{2} = 4} \implies y = 10$$

So, answer is Option $(C)$.

Answer 2

The perpendicular distance between (1,2) is same as (5,10) but of opposite sign.