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Chapter 18: Problem 36

The point \((2,3)\) undergoes the following three transformations successively (i) reflection about the line \(y=x\) (ii) translation through a distance 2 units along the positive direction of \(y\)-axis (iii) rotation through an angle of \(45^{\circ}\) about the origin in the anti- clockwise direction. The final coordinates of the point are (A) \(\left(\frac{1}{\sqrt{2}}, \frac{7}{\sqrt{2}}\right)\) (B) \(\left(-\frac{1}{\sqrt{2}}, \frac{7}{\sqrt{2}}\right)\) (C) \(\left(\frac{1}{\sqrt{2}},-\frac{7}{\sqrt{2}}\right)\) (D) none of these

Short Answer

Expert verified
The final coordinates are (-1/鈭�2, 7/鈭�2).

Step by step solution

01

Reflect the point about the line y = x

To reflect the point \((2,3)\) about the line \(y = x\), we swap the coordinates. So, the reflection of \((2,3)\) is \((3,2)\).
02

Translate the point along the positive y-axis

Next, we translate the reflected point \((3, 2)\) by 2 units along the positive y-axis. This involves adding 2 to the y-coordinate. Thus, the translated point is \((3, 4)\).
03

Rotate the point 45 degrees anticlockwise about the origin

To rotate the point \((3, 4)\) 45 degrees anticlockwise about the origin, we use the rotation formulas: \(x' = x \cos(45^{\circ}) - y \sin(45^{\circ})\) and \(y' = x \sin(45^{\circ}) + y \cos(45^{\circ})\).\ Substituting the values, we use \(\cos(45^{\circ}) = \sin(45^{\circ}) = \frac{1}{\sqrt{2}}\):\ \[x' = 3 \cdot \frac{1}{\sqrt{2}} - 4 \cdot \frac{1}{\sqrt{2}} = \frac{3}{\sqrt{2}} - \frac{4}{\sqrt{2}} = -\frac{1}{\sqrt{2}}\] \[y' = 3 \cdot \frac{1}{\sqrt{2}} + 4 \cdot \frac{1}{\sqrt{2}} = \frac{3}{\sqrt{2}} + \frac{4}{\sqrt{2}} = \frac{7}{\sqrt{2}}\]\ Thus, the rotated point is \(\left(-\frac{1}{\sqrt{2}}, \frac{7}{\sqrt{2}}\right)\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Reflection Transformation
Reflection transformation in coordinate geometry is a way of mirroring a point across a specific line. Imagine it like looking in a mirror and seeing your reflection on the other side. When reflecting a point
  • Across the x-axis: Only the y-coordinate changes its sign.
  • Across the y-axis: Only the x-coordinate changes its sign.
  • Across the line y = x: The x and y coordinates swap places.
For example, when we reflect the point \(2,3\) across the line \(y=x\), we switch the coordinates to get \(3,2\). This concept is crucial because it gives us symmetry in geometry, allowing us to predict positions after a reflection transformation.
Translation in Geometry
Translation in geometry refers to sliding a point or a shape to a new location without altering its orientation. Picture it as pushing an object along a table without rotating it. You move every point by the same distance in the same direction.
For translating:
  • Along the x-axis: Add or subtract a certain value to the x-coordinate.
  • Along the y-axis: Add or subtract a certain value to the y-coordinate.
In our exercise, after reflecting the point \(3,2\), we translated it 2 units up along the y-axis, resulting in \(3,4\). This translation step increases the y-coordinate by 2, leaving the x-coordinate unchanged. Understanding translation helps visualize how objects move across a plane without changing their appearance.
Rotation Transformation
Rotation transformation involves turning a point or a shape around a central point, usually the origin in coordinate geometry. Imagine spinning a piece of paper around a pin stuck through its center. The points move but stay equidistant from the center.
For rotation:
  • Use angles measured in degrees or radians, with a common example being 45掳 or \(\pi/4\) radians.
  • The rotation formulas are \(x' = x\cos\theta - y\sin\theta\) and \(y' = x\sin\theta + y\cos\theta\).
In our scenario, using an angle of 45掳, the point \(3,4\) rotates around the origin to become \(-\frac{1}{\sqrt{2}}, \frac{7}{\sqrt{2}}\). These transformations are essential for understanding how objects can be manipulated in space, crucial for both theoretical mathematics and practical applications such as graphics design or engineering.

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Most popular questions from this chapter

The equation of the directrix of the parabola \(y^{2}+4 y+\) \(4 x+2=0\) is: (A) \(x=-1\) (B) \(x=1\) (C) \(x=-\frac{3}{2}\) (D) \(x=\frac{3}{2}\)

In oblique coordinates, the equation \(y=m x+c\) represents a straight line which is inclined at an angle $$ \tan ^{-1}\left(\frac{m \sin w}{1+m \cos w}\right) $$ to the \(x\)-axis, where \(w\) is the angle between the axes. If \(\theta\) be the angle between two lines \(y=m_{1} x+c_{1}\) and \(y=m_{2} x\) \(+c_{2}, w\) be the angle between the axes, then $$ \tan \theta=\frac{\left(m_{1}-m_{2}\right) \sin w}{1+\left(m_{1}+m_{2}\right) \cos w+m_{1} m_{2}} $$ The two given lines are parallel if \(m_{1}=m_{2}\). The two lines are perpendicular if \(1+\left(m_{1}+m_{2}\right) \cos w+\) \(m_{1} m_{2}=0\) If \(y=x \tan \frac{11 \pi}{24}\) and \(y=x \tan \frac{19 \pi}{24}\) represent two straight lines at right angles, then the angle between the axes is (A) \(\frac{\pi}{6}\) (B) \(\frac{\pi}{4}\) (C) \(\frac{\pi}{3}\) (D) \(\frac{\pi}{2}\)

If \(x_{1}, x_{2}, x_{3}\) as well as \(y_{1}, y_{2}, y_{3}\) are in G. P. with the same common ratio, then the points \(\left(x_{1}, y_{1}\right),\left(x_{2}, y_{2}\right)\) and \(\left(x_{3}, y_{3}\right)\) (A) lie on a straight line (B) lie on an ellipse (C) lie on a circle (D) are vertices of a triangle

If a vertex of a triangle is \((1,1)\) and the mid-points of two sides through this vertex are \((-1,2)\) and \((3,2)\) then the centroid of the triangle is (A) \(\left(-1, \frac{7}{3}\right)\) (B) \(\left(\frac{-1}{3}, \frac{7}{3}\right)\) (C) \(\left(1, \frac{7}{3}\right)\) (D) \(\left(\frac{1}{3}, \frac{7}{3}\right)\)

Two points \(A\) and \(B\) are given. \(P\) is a moving point on one side of the line \(A B\) such that \(\angle P A B-\angle P B A\) is a positive constant \(2 \theta\). The locus of the point \(P\) is (A) \(x^{2}+y^{2}+2 x y \cot 2 \theta=a^{2}\) (B) \(x^{2}+y^{2}-2 x y \cot 2 \theta=a^{2}\) (C) \(x^{2}+y^{2}+2 x y \tan 2 \theta=a^{2}\) (D) \(x^{2}-y^{2}+2 x y \cot 2 \theta=a^{2}\).
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