If the image of point $P(2, 3)$ in a line $L$ is $Q(4,5)$, then the image of point $R(0,0)$ in the same line is:

Question

If $ x = a(\cos \theta + \theta \sin \theta), \, y = a(\sin \theta - \theta \cos \theta) $, then $ \frac{d^2 y}{dx^2} $ equals

Answer 1

To solve this problem, we need to find the equation of the line $L$ given the image of point $P(2, 3)$ is $Q(4,5)$ and use that line to determine the image of another point $R(0,0)$ in the same line.

The key concept here involves the properties of reflection across a line. If a point $(x_1, y_1)$ is reflected across a line to another point $(x_2, y_2)$, the midpoint of that line segment will lie on the line. Therefore, the midpoint $(x_m, y_m)$ can be found by the formula:

$x_m = \frac{x_1 + x_2}{2}$, $y_m = \frac{y_1 + y_2}{2}$

Substituting the given points $P(2, 3)$ and $Q(4,5)$ into the midpoint formula gives us:

$x_m = \frac{2 + 4}{2} = 3$

$y_m = \frac{3 + 5}{2} = 4$

Thus, the midpoint $(3, 4)$ lies on the line $L$.

For the line $L$, assume it has the standard form $ax + by + c = 0$. Using the midpoint condition, the equation must satisfy $3a + 4b + c = 0$.

Since $P$ and $Q$ are images of each other, the line $L$ is perpendicular to the line segment joining $P$ and $Q$. The slope of $PQ$ is 1 (since slope $= \frac{5-3}{4-2} = 1$), thus, the slope of the line $L$ is $-1$.

This gives the equation of $L$ as $x + y = 7$, using the point $(3,4)$ to determine the constant term.

With the line equation, find the image of the point $(0,0)$:

Use the formula for finding the image of a point $(x_1, y_1)$ with respect to the line $ax + by + c = 0$:

$x' = \frac{x_1 (b^2 - a^2) - 2y_1 ab - 2ac}{a^2 + b^2}$

$y' = \frac{y_1 (a^2 - b^2) - 2x_1 ab - 2bc}{a^2 + b^2}$

For the line $x + y = 7$, we can express it as $a = 1, b = 1, c = -7$.

Substitute $(x_1, y_1) = (0,0)$ into the formula:

\[x' = \frac{0(1 - 1) - 0 - 2(-7)}{1 + 1} = \frac{14}{2} = 7 \] \[ y' = \frac{0(1 - 1) - 0 - 2(-7)}{1 + 1} = \frac{14}{2} = 7\]

Therefore, the image of the point $(0,0)$ is $(7,7)$.

If the image of point $P(2, 3)$ in a line $L$ is $Q(4,5)$, then the image of point $R(0,0)$ in the same line is:

The Correct Option is D

Solution and Explanation

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