Let the line x + y = 1$x + y = 1$ meet the axes of x$x$ and y$y$ at A and B, respectively. A right angled triangle AMN is inscribed in the triangle OAB, where O is the origin and the points M and N lie on the lines OB and AB, respectively. If the area of the triangle AMN is (4)/(9)$\frac{4}{9}$ of the area of the triangle OAB and AN: NB = \lambda : 1, then the sum of all possible value(s) of λ$\lambda$ is :

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Core Logic

Decode slope angle fields to define vector shifts. Match variables to isolate focal root segments accurately as provided in the reference layout steps.

Step 1: Identify Phase Shift Angles

Given final tracking position slope component is:

Because clockwise translation turns angular coordinates down by half increments:

Step 2: Construct Rotated Vector Path Line

The line equation through point (a,0)$(a,0)$ with final angle scaling is:

Step 3: Solve Origin Geometric Profiles

Impose perpendicular bounds tracking distance rule:

Target formulation output text string solution:

{{SOL_IMG_62}}

Pattern Recognition

Recognizing standard trigonometric slopes (15^° = 2-√(3)$\tan 15^\circ = 2-\sqrt{3}$) bypasses computational bottlenecks instantly when working with line updates.

Chapter Mix

Class 11 Mathematics: Straight Lines

Related Formula

Core Logic

Identify which diagonal passes through the origin. Use its directional configuration slope to determine the exact angle orientation layout map of the geometric system elements.

Step 1: Determine Diagonal Orientation

The line passing through the origin is diagonal OB$OB$:

Since diagonals of a square split the vertex paths evenly at 45^°$45^\circ$, the side OA$OA$ makes an angle:

Step 2: Coordinate Vector Definition

Coordinates of vertex A$A$ matching side value a$a$ are:

Step 3: Diagonal Intersection Point Solver

Vertex A$A$ lies upon the alternative structural path line:

a² = 48$a^2 = 48$

{{SOL_IMG_57}}

Pattern Recognition

Slopes involving √(3) ± 1$\sqrt{3} \pm 1$ explicitly alert you to geometric orientations built on multiples of 15^circ$15^circ$ or 75^circ$75^circ$ configurations.

Chapter Mix

Class 11 Mathematics: Straight Lines

Related Formula

Perpendicular distance from a point (x₁, y₁)$(x_1, y_1)$ to a line Ax + By + C = 0$Ax + By + C = 0$:

Core Logic

Rewrite the summation for α$\alpha$ by embedding complex roots to isolate alternating terms:

Using complex cube roots of unity conversions (1 + √(3)i = -2ω²$1 + \sqrt{3}i = -2\omega^2$), this simplifies directly to α = 1$\alpha = 1$.

Step 1: Calculating Perpendicular Distance

Substitute α = 1$\alpha = 1$ into the line equation, yielding x - √(3)y + 1 = 0$x - \sqrt{3}y + 1 = 0$.

Evaluating the distance from point (12, √(3))$(12, \sqrt{3})$:

Pattern Recognition

Binomial series containing terms like √(3)2r-1$\sqrt{3}^{2r-1}$ collapse neatly into simple geometric forms using standard complex rotations (eiπ/3$e^{i\pi/3}$).

Chapter Mix

Class 11 Maths: Binomial Theorem Class 11 Maths: Straight Lines

Related Formula

Slope angle interaction tracking: In right triangle AMB$\triangle AMB$, the tangent of inclination satisfies:

Core Logic

A line tracking equal angles through positive coordinate frames has equation y=x$y=x$ [cite: 1353]. Its slope parameter is m₁ = 1$m_1 = 1$ [cite: 1353].

The parallel boundary lines L₁$L_1$ and L₂$L_2$ have a uniform slope value of m₂ = -2$m_2 = -2$ [cite: 1353].

Let θ$\theta$ represent the precise geometric crossing angle matching the line y=x$y=x$ intersecting line L₂$L_2$[cite: 1353].

Step 1: Finding the requested ratio

Using the slope interaction formula to calculate the tangent angle metric[cite: 1353]: θ = | (m₁ - m₂)/(1 + m₁ m₂) |$$\tan \theta = \left| \frac{m_1 - m_2}{1 + m_1 m_2} \right|$$ [cite: 1353] θ = | (1 - (-2))/(1 + (1)(-2)) | = | (3)/(-1) | = 3$$\tan \theta = \left| \frac{1 - (-2)}{1 + (1)(-2)} \right| = \left| \frac{3}{-1} \right| = 3$$ [cite: 1353]

By observing right-angled geometry configuration AMB$\triangle AMB$ [cite: 1353]: (AM)/(BM) = θ = 3$$\frac{AM}{BM} = \tan \theta = 3$$ [cite: 1353]

Pattern Recognition

Ratios of side projections of lines intersecting parallel setups depend completely on direction orientations, bypassing raw variable solving steps entirely.

Chapter Mix

Class 11 Mathematics: Straight Lines