Question

Suppose the point \(P(1,1)\) is translated to \(Q\) in the direction of y = 2x. If \(PQ = 1\),then Qis

• A. \((2, 0)\)
• B. \((0,2)\)
• C. \((\dfrac{√2+1}{√2},\dfrac{√2+1}{√2})\)
• D. \((\dfrac{√5+1}{√5},\dfrac{√5+2}{√5})\)
• E. \((\dfrac{2+√3}{2},\dfrac{3}{2})\)

Answer 1

The Correct Option is D

We need to find the new position \( Q \) of the point \( P(1, 1) \) after translating it a distance of 1 unit in the direction of the line \( y = 2x \).

Step 1: Determine the direction vector.

The line \( y = 2x \) has a slope of 2, so its direction vector is \( \vec{v} = (1, 2) \). To get a unit vector in this direction, we normalize \( \vec{v} \):

\[ \|\vec{v}\| = \sqrt{1^2 + 2^2} = \sqrt{5} \] \[ \hat{v} = \left( \frac{1}{\sqrt{5}}, \frac{2}{\sqrt{5}} \right) \]

Step 2: Compute the translation.

Translating \( P(1, 1) \) by 1 unit in the direction of \( \hat{v} \) gives:

\[ Q = P + 1 \cdot \hat{v} = \left( 1 + \frac{1}{\sqrt{5}}, 1 + \frac{2}{\sqrt{5}} \right) = \left( \frac{\sqrt{5} + 1}{\sqrt{5}}, \frac{\sqrt{5} + 2}{\sqrt{5}} \right) \]

Step 3: Verify the distance.

The distance between \( P \) and \( Q \) is:

\[ PQ = \sqrt{ \left( \frac{1}{\sqrt{5}} \right)^2 + \left( \frac{2}{\sqrt{5}} \right)^2 } = \sqrt{ \frac{1}{5} + \frac{4}{5} } = \sqrt{1} = 1 \]

This confirms that the translation is correct.

Step 4: Match with the options.

The coordinates of \( Q \) match option (D):

\[ \left( \frac{\sqrt{5} + 1}{\sqrt{5}}, \frac{\sqrt{5} + 2}{\sqrt{5}} \right) \]

Answer 2

Step 1: Find the direction vector of the line \( y = 2x \).

The line \( y = 2x \) has a slope of 2. A direction vector for this line is:

\[ \vec{v} = (1, 2) \]

To normalize this vector, compute its magnitude:

\[ |\vec{v}| = \sqrt{1^2 + 2^2} = \sqrt{5} \]

The unit vector in the direction of \( \vec{v} \) is:

\[ \hat{v} = \left( \frac{1}{\sqrt{5}}, \frac{2}{\sqrt{5}} \right) \]

Step 2: Translate \( P(1,1) \) by 1 unit in the direction of \( \hat{v} \).

The coordinates of \( Q \) are obtained by adding the scaled unit vector \( \hat{v} \) to \( P(1,1) \):

\[ Q_x = 1 + \frac{1}{\sqrt{5}}, \quad Q_y = 1 + \frac{2}{\sqrt{5}} \]

Thus, the coordinates of \( Q \) are:

\[ Q = \left( 1 + \frac{1}{\sqrt{5}}, 1 + \frac{2}{\sqrt{5}} \right) \]

Step 3: Simplify the coordinates of \( Q \).

Rewriting \( Q \) in a compact form:

\[ Q = \left( \frac{\sqrt{5} + 1}{\sqrt{5}}, \frac{\sqrt{5} + 2}{\sqrt{5}} \right) \]

Step 4: Match with the given options.

The coordinates of \( Q \) match option \( D \):

\[ \left( \frac{\sqrt{5} + 1}{\sqrt{5}}, \frac{\sqrt{5} + 2}{\sqrt{5}} \right) \]

Final Answer:

\( \left( \frac{\sqrt{5} + 1}{\sqrt{5}}, \frac{\sqrt{5} + 2}{\sqrt{5}} \right) \)