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 :

Related Formula

Core Logic

Let the point on the line be Q$Q$. Using the parametric form of a line, the coordinates of Q$Q$ at a distance r = √((2)/(3))$r = \sqrt{\frac{2}{3}}$ from P(2,3)$P(2,3)$ are:

Since Q$Q$ lies on the line x + y = 6$x + y = 6$, substitute these coordinates into the equation.

Step 1: Solving the Trigonometric Equation

Step 2: Squaring to find angles

Square both sides:

The general solutions for 2θ$2\theta$ in [0, 2π]$[0, 2\pi]$ are (π)/(6)$\frac{\pi}{6}$ and (5π)/(6)$\frac{5\pi}{6}$.

Therefore, θ₁ = (π)/(12)$\theta_1 = \frac{\pi}{12}$ and θ₂ = (5π)/(12)$\theta_2 = \frac{5\pi}{12}$.

Pattern Recognition

Whenever distance r$r$ from a fixed point to a line is given along with a variable angle, the parametric coordinates (x₁ + r θ, y₁ + r θ)$(x_1 + r\cos\theta, y_1 + r\sin\theta)$ substitute cleanly into the target line equation to produce a standard trigonometric identity.

Chapter Mix

Class 11 Maths: Straight Lines Class 11 Maths: Trigonometric Equations

Core Logic

Step 1: Distance Mapping

The perpendicular distance from O(0,0)$O(0,0)$ to side BC$BC$ is OD$OD$:

Since O$O$ is the centroid, it divides the altitude AD$AD$ in a 2:1$2:1$ ratio.

Total altitude length AD = (8)/(3) + (4)/(3) = 4$AD = \frac{8}{3} + \frac{4}{3} = 4$.

Step 2: Solve for Coordinates

The distance from A(α, β)$A(\alpha, \beta)$ to the line BC$BC$ is the altitude AD$AD$:

Substitute β = 2√(2)α$\beta = 2\sqrt{2}\alpha$:

Since A$A$ and the origin O$O$ must lie on opposite sides of BC$BC$ (wait, O$O$ is inside the triangle, so A$A$ and O$O$ lie on opposite sides of BC$BC$? No, O$O$ is inside the triangle, so the origin and vertex A are on opposite sides of the chord BC$BC$ if we look from the circumcenter. Wait, O(0,0)$O(0,0)$ gives 0+0-4 = -4 < 0$0+0-4 = -4 < 0$. If α = 16/9, β = 32√(2)/9$\alpha = 16/9, \beta = 32\sqrt{2}/9$, then 16/9 + 2√(2)(32√(2)/9) - 4 = 16/9 + 128/9 - 36/9 = 108/9 = 12 > 0$16/9 + 2\sqrt{2}(32\sqrt{2}/9) - 4 = 16/9 + 128/9 - 36/9 = 108/9 = 12 > 0$. Thus, they lie on opposite sides, which is correct for altitude line. Wait, our source notes: A(α,β)$A(\alpha,\beta)$ and (0,0)$(0,0)$ lie on the SAME side of the given line is Rejected. Actually, they lie on opposite sides relative to BC$BC$. Thus (α, β) = (-(8)/(9), -16√(2)9)$(\alpha, \beta) = \left(-\frac{8}{9}, \frac{-16\sqrt{2}}{9}\right)$ is correct because O$O$ is the centroid, so moving from D$D$ to O$O$ and then to A$A$ implies O$O$ is between A$A$ and D$D$. Let's trust the solved matrix: A(α, β) = (-(8)/(9), -16√(2)9)$A(\alpha, \beta) = \left(-\frac{8}{9}, \frac{-16\sqrt{2}}{9}\right)$.

Step 3: Final Value Evaluation

We need the greatest integer less than or equal to |α + √(2)β|$|\alpha + \sqrt{2}\beta|$:

[4.44] = 4$[4.44] = 4$

Chapter Mix

Class 11 Mathematics: Straight Lines

Related Formula

Core Logic

We write down the normal equation of the straight line using the given polar normal angle α = 45^°$\alpha = 45^\circ$, find its intercept coordinates, and use the area constraint to solve for the coefficients.

Step 1: Write down normal form

The perpendicular drawn from the origin makes an angle of 45^°$45^\circ$ with the positive x$x$-axis, so α = 45^°$\alpha = 45^\circ$. The line equation is:

Comparing this with the given format x + by + c = 0$x + by + c = 0$, we find:

Step 2: Solve for the parameters using the area constraint

The line equation is x + y = p√(2)$x + y = p\sqrt{2}$. The intercepts are:

• xintercept = p√(2)$x_{\text{intercept}} = p\sqrt{2}$
• yintercept = p√(2)$y_{\text{intercept}} = p\sqrt{2}$

The area of the right-angled triangle formed with the axes is:

Since the area is given as 48 square units:

p² = 48$p^2 = 48$

Step 3: Calculate the requested value

We have:

b² = 1² = 1$b^2 = 1^2 = 1$

Therefore, we find:

Pattern Recognition

Normal equation coupling: Normal equations of lines x α+y α = p$x\cos\alpha+y\sin\alpha = p$ are extremely powerful when normal angles are specified. For α=45^°$\alpha=45^\circ$, the coordinate intercepts are identical, making the area relation A=p²$A=p^2$ exceptionally simple.

Chapter Mix

Class 11 Mathematics: Straight Lines

Related Formula

A family of lines passing through the intersection of L₁ = 0$L_1 = 0$ and L₂ = 0$L_2 = 0$ is expressed as:

For a point P$P$ through which a family of lines passes, the line in the family that is at the maximum distance from origin O$O$ is the line perpendicular to OP$OP$ passing through P$P$.

Core Logic

Rearranging the equation of the given lines in terms of λ$\lambda$:

To find point P$P$, solve the system:

• x + 2y = 5 x = 5 - 2y$x + 2y = 5 \implies x = 5 - 2y$
• 3x + 7y = 17$3x + 7y = 17$

Step 1: Finding P$P$ and Line L$L$

Substitute x = 5-2y$x = 5-2y$ into the second equation:

Thus, the common intersection point is P(1, 2)$P(1, 2)$.

The line farthest from the origin is perpendicular to the segment OP$OP$ joining the origin O(0,0)$O(0,0)$ to P(1,2)$P(1,2)$.

• Slope of OP = (2 - 0)/(1 - 0) = 2$OP = \frac{2 - 0}{1 - 0} = 2$
• Slope of L$L$ (m$m$) = -(1)/(2)$= -\frac{1}{2}$

Equation of line L$L$ passing through P(1,2)$P(1,2)$:

Step 2: Distance calculation from (3,6)$(3,6)$

The distance d$d$ of point Q(3,6)$Q(3,6)$ from line x + 2y - 5 = 0$x + 2y - 5 = 0$ is:

Calculating d²$d^2$:

Pattern Recognition

Short Shortcut: The maximum distance of a family of lines passing through P$P$ from the origin is simply the length OP$OP$. The line perpendicular to OP$OP$ at P$P$ is the unique farthest line.

Chapter Mix

Class 11 Mathematics: Straight Lines

Related Formula

The orthocentre P$P$ of a triangle is the point of intersection of its altitudes. Area of a triangle with a horizontal base lying on the x-axis is:

Core Logic

Find vertex A$A$ by solving the line equations AB$AB$ and AC$AC$:

Substitute into x + y = 2 (3y - 2) + y = 2 4y = 4 y = 1$x + y = 2 \implies (3y - 2) + y = 2 \implies 4y = 4 \implies y = 1$. Then x = 3(1) - 2 = 1$x = 3(1) - 2 = 1$. So vertex A$A$ is (1, 1)$(1, 1)$.

Find vertices B$B$ and C$C$ where the lines cross the x-axis (y = 0$y = 0$):

• For B$B$ (on line AB$AB$): 3(0) - x = 2 x = -2 B(-2, 0)$3(0) - x = 2 \implies x = -2 \implies B(-2, 0)$
• For C$C$ (on line AC$AC$): x + 0 = 2 x = 2 C(2, 0)$x + 0 = 2 \implies x = 2 \implies C(2, 0)$

Base length BC = |2 - (-2)| = 4$BC = |2 - (-2)| = 4$.

Step 1: Find Equations of Altitudes

• Altitude from A to BC:

Since BC$BC$ lies along the x-axis, the altitude from A(1, 1)$A(1, 1)$ must be a vertical line:

• Altitude from B to AC:

Slope of line AC$AC$ (x + y = 2$x + y = 2$) is mAC = -1$m_{AC} = -1$. Therefore, the slope of the altitude perpendicular to AC$AC$ is m₂ = -(1)/(-1) = 1$m_2 = -\frac{1}{-1} = 1$. Passing through B(-2, 0)$B(-2, 0)$:

Step 2: Solve for Orthocentre coordinates P

Intersect the altitude equations: x = 1$x = 1$ and y = x + 2$y = x + 2$: y = 1 + 2 = 3$y = 1 + 2 = 3$

Hence, the orthocentre is P(1, 3)$P(1, 3)$.

Step 3: Compute Area of Triangle PBC

Triangle PBC$PBC$ has base BC = 4$BC = 4$ on the x-axis, and vertex P(1, 3)$P(1, 3)$ gives a height of 3$3$.

Pattern Recognition

When a triangle has its base sitting entirely on the coordinate axis, the altitude dropped from the opposite vertex is simplified to a pure horizontal or vertical line path, heavily reducing your linear derivation equation workload.

Chapter Mix

Class 11 Mathematics: Straight Lines