The point $P(a, b)$ undergoes the following three transformations successively: (a) reflection about the line $y = x$. (b) translation through 2 units along the positive direction of x-axis. (c) rotation through angle $\frac{\pi}{4}$ about the origin in the anti-clockwise direction. If the co-ordinates of the final position of the point $P$ are $(-\frac{1}{\sqrt{2}}, \frac{7}{\sqrt{2}})$, then the value of $2a + b$ is equal to : Options: (A) 13 (B) 9 (C) 5 (D) 7

### Analyzing the Setup We begin with a point $P(a, b)$ on the Cartesian plane. Our goal is to track its transformation through three distinct geometric operations to determine the values of $a$ and $b$. ### Phase 1: The Mirror The journey begins with a reflection about the line $y = x$. This transformation swaps the $x$ and $y$ coordinates of the point. Thus, the point $P(a, b)$ transforms into $P'(b, a)$. ### Phase 2: The Shift Next, we apply a translation of $2$ units along the positive $x$-axis. This horizontal slide increases the $x$-coordinate by $2$ while leaving the $y$-coordinate unchanged. The point $P'$ evolves into $P''(b + 2, a)$. ### Phase 3: The Spin We now rotate the point $P''(b + 2, a)$ by an angle $\theta = \frac{\pi}{4}$ anti-clockwise about the origin. The rotation transformation is defined by: $$X_{new} = x \cos \theta - y \sin \theta$$ $$Y_{new} = x \sin \theta + y \cos \theta$$ Substituting $x = b + 2$, $y = a$, and $\theta = \frac{\pi}{4}$, and noting that $\cos(\frac{\pi}{4}) = \sin(\frac{\pi}{4}) = \frac{1}{\sqrt{2}}$, we obtain: $$X_{new} = \frac{b + 2 - a}{\sqrt{2}}$$ $$Y_{new} = \frac{b + 2 + a}{\sqrt{2}}$$ ### Phase 4: The Resolution We are given that the final coordinates are $(-\frac{1}{\sqrt{2}}, \frac{7}{\sqrt{2}})$. Equating our expressions to these values, we establish the following system of equations: $$\frac{b - a + 2}{\sqrt{2}} = -\frac{1}{\sqrt{2}} \Rightarrow b - a = -3$$ $$\frac{b + a + 2}{\sqrt{2}} = \frac{7}{\sqrt{2}} \Rightarrow b + a = 5$$ Adding these two equations, we find $2b = 2$, which implies $b = 1$. Substituting $b = 1$ into the second equation, we find $1 + a = 5$, which implies $a = 4$. ### Final Calculation The problem asks for the value of $2a + b$. Substituting our derived values: $$2(4) + 1 = 8 + 1 = 9$$ The final result is **9**.

I always forget the signs in the rotation formula. Is it $x \cos \theta + y \sin \theta$ or minus?

It is a common pitfall! Think of the unit circle. If you rotate a point $(1, 0)$ by $\theta$, it lands at $(\cos \theta, \sin \theta)$. If you rotate $(0, 1)$ by $\theta$, it lands at $(-\sin \theta, \cos \theta)$. Using the linearity of rotation, the new $x$ is $x \cos \theta - y \sin \theta$ and the new $y$ is $x \sin \theta + y \cos \theta$. Visualizing the unit vectors is the best way to never forget this.

JEE Main 2021 (27 July Shift 2)

LEVELJEE Main

Animated Solution for Mathematics - Straight Lines: The point $P (a, b)$ undergoes the following three transformations successively: (a) reflection about the line $y = x$ . (b) translation through 2 units along the positive direction of x-axis. (c) rotation through angle $\frac{π}{4}$ about the origin in the anti-clockwise direction. If the co-ordinates of the final position of the point $P$ are $(- \frac{1}{2}, \frac{7}{2})$ , then the value of $2 a + b$ is equal to :

Select Answer:

Progressive Hints

Hint 1

Visualize the point

P (a, b)

as a dancer on a stage. The key to this problem is the sequence. Transformations are like a recipe; if you change the order, the dish changes. Do not try to combine them into one giant matrix yet. Take it one step at a time: reflect, then shift, then spin.

Hint 2

For the final step, you need the rotation transformation. Recall that rotating a point

(x, y)

by an angle

θ

anti-clockwise about the origin results in new coordinates

(x^{'}, y^{'})

given by

x^{'} = x cos θ - y sin θ

and

y^{'} = x sin θ + y cos θ

. Keep this formula etched in your memory.

Hint 3

Start by applying the reflection:

(a, b) \to (b, a)

. Then, apply the translation:

(b, a) \to (b + 2, a)

. Finally, plug

(b + 2, a)

into the rotation formulas with

θ = \frac{π}{4}

. Set these equal to the given final coordinates

(- \frac{1}{2}, \frac{7}{2})

to generate your system of linear equations.

Visualized Solution

The Journey of Point $P (a, b)$

Initial point: $P (a, b)$
Goal: Find $2 a + b$ after three successive transformations.

Transformation 1: Reflection

Step 1: Reflection about the line $y = x$
Rule: $(x, y) \to (y, x)$
New coordinates: $P^{'} (b, a)$

Transformation 2: Translation

Step 2: Translation by $+ 2$ units along the X-axis.
Rule: $(x, y) \to (x + 2, y)$
New coordinates: $P^{''} (b + 2, a)$

Transformation 3: Rotation Formula

Step 3: Rotation by $θ = \frac{π}{4}$ anti-clockwise about the origin.
Standard Rotation Formula:
$X_{n e w} = x cos θ - y sin θ$
$Y_{n e w} = x sin θ + y cos θ$

Substituting into Rotation Formula

Substitute $(x, y) = (b + 2, a)$ and $θ = \frac{π}{4}$
$X_{n e w} = (b + 2) cos (\frac{π}{4}) - a sin (\frac{π}{4})$
$Y_{n e w} = (b + 2) sin (\frac{π}{4}) + a cos (\frac{π}{4})$

Evaluating Trigonometric Values

We know $cos (\frac{π}{4}) = sin (\frac{π}{4}) = \frac{1}{2}$
$X_{n e w} = \frac{b + 2 - a}{2}$
$Y_{n e w} = \frac{b + 2 + a}{2}$

Equating the X-Coordinates

Given final position: $(- \frac{1}{2}, \frac{7}{2})$
Equating X-coordinates:
$\frac{b - a + 2}{2} = - \frac{1}{2}$
$b - a + 2 = - 1$
Equation 1: $b - a = - 3$

Equating the Y-Coordinates

Equating Y-coordinates:
$\frac{b + a + 2}{2} = \frac{7}{2}$
$b + a + 2 = 7$
Equation 2: $b + a = 5$

Solving for $a$ and $b$

System of equations:
$b - a = - 3$
$b + a = 5$
Adding both equations: $2 b = 2 \Rightarrow b = 1$
Substituting $b = 1$ gives $a = 4$

Final Target Calculation

We found: $a = 4$ , $b = 1$
Target expression: $2 a + b$
$2 (4) + 1 = 8 + 1 = 9$
Final Answer: 9

00:00 / 00:00

The Sigma Insight: Distance and Section Formulas

Solution Diagram

Analyzing the Setup

We begin with a point

P (a, b)

on the Cartesian plane. Our goal is to track its transformation through three distinct geometric operations to determine the values of

a

and

b

Phase 1

The Mirror

The journey begins with a reflection about the line

y = x

. This transformation swaps the

x

and

y

coordinates of the point.

Thus, the point

P (a, b)

transforms into

P^{'} (b, a)

Phase 2

The Shift

Next, we apply a translation of

2

units along the positive

x

-axis. This horizontal slide increases the

x

-coordinate by

2

while leaving the

y

-coordinate unchanged.

The point

P^{'}

evolves into

P^{''} (b + 2, a)

Phase 3

The Spin

We now rotate the point

P^{''} (b + 2, a)

by an angle

θ = \frac{π}{4}

anti-clockwise about the origin. The rotation transformation is defined by:

X_{n e w} = x cos θ - y sin θ

Y_{n e w} = x sin θ + y cos θ

Substituting

x = b + 2

y = a

, and

θ = \frac{π}{4}

, and noting that

cos (\frac{π}{4}) = sin (\frac{π}{4}) = \frac{1}{2}

, we obtain:

X_{n e w} = \frac{b + 2 - a}{2}

Y_{n e w} = \frac{b + 2 + a}{2}

Phase 4

The Resolution

We are given that the final coordinates are

(- \frac{1}{2}, \frac{7}{2})

. Equating our expressions to these values, we establish the following system of equations:

\frac{b - a + 2}{2} = - \frac{1}{2} \Rightarrow b - a = - 3

\frac{b + a + 2}{2} = \frac{7}{2} \Rightarrow b + a = 5

Adding these two equations, we find

2 b = 2

, which implies

b = 1

. Substituting

b = 1

into the second equation, we find

1 + a = 5

, which implies

a = 4

Final Calculation

The problem asks for the value of

2 a + b

. Substituting our derived values:

2 (4) + 1 = 8 + 1 = 9

The final result is 9.

Common Traps & FAQs

Wait, does the order of transformations really matter? Can't I just rotate first and then reflect?

Excellent question! In the world of linear algebra, transformations are generally non-commutative. Think of it like putting on socks and shoes—the order dictates the outcome. If you rotate first, the line of reflection

y = x

also rotates, changing the entire problem. Always follow the sequence strictly as given in the problem statement.

I always forget the signs in the rotation formula. Is it

x cos θ + y sin θ

or minus?

It is a common pitfall! Think of the unit circle. If you rotate a point

(1, 0)

θ

, it lands at

(cos θ, sin θ)

. If you rotate

(0, 1)

θ

, it lands at

(- sin θ, cos θ)

. Using the linearity of rotation, the new

x

x cos θ - y sin θ

and the new

y

x sin θ + y cos θ

. Visualizing the unit vectors is the best way to never forget this.

Is there a faster way to do this without the rotation formula?

Yes! Complex numbers are your secret weapon. Represent the point

(x, y)

z = x + i y

. A rotation by

θ

is simply multiplying by

e^{i θ}

. So,

z_{f ina l} = (z_{a f t er_t r an s l a t i o n}) \cdot e^{iπ /4}

. It is elegant, but for this specific problem, the standard rotation formula is often safer and less prone to calculation errors.

Select Answer:

Visualized Solution

The Journey of Point $P (a, b)$

Transformation 1: Reflection

Transformation 2: Translation

Transformation 3: Rotation Formula

Substituting into Rotation Formula

Evaluating Trigonometric Values

Equating the X-Coordinates

Equating the Y-Coordinates

Solving for $a$ and $b$

Final Target Calculation

The Sigma Insight: Distance and Section Formulas

Analyzing the Setup

Phase 1

Phase 2

Phase 3

Phase 4

Final Calculation

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Select Answer:

Visualized Solution

The Journey of Point .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } P(a,b)

Transformation 1: Reflection

Transformation 2: Translation

Transformation 3: Rotation Formula

Substituting into Rotation Formula

Evaluating Trigonometric Values

Equating the X-Coordinates

Equating the Y-Coordinates

Solving for .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } a and .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } b

Final Target Calculation

The Sigma Insight: Distance and Section Formulas

Analyzing the Setup

Phase 1

Phase 2

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Final Calculation

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