Straight Lines
Straight Line
Allen Star Batch
Grade 11
Question:
If a line is passing through the point $P(1,2)$ cutting the lines $x + y - 5 = 0$ and $2x - y - 7$ at $A$ and $B$ respectively such that the harmonic mean of $PA$ and $PB$ is $10$. If equation of line is $(y - 2) = \tan\left[\pi - \sin^{-1}\left(\frac{a}{\sqrt{146}}\right) - \sin^{-1}\left(\frac{b}{\sqrt{146}}\right)\right](x - 1)$. The $|a - b|\frac{25}{41}$ is ______.
Step-by-Step Solution
Key Concept: Parametric representation with direction angles combined with line constraints and harmonic mean conditions reduces the problem to solving a linear trigonometric equation.
For the line passing through $P(1,2)$ with slope $m$, points $A$ and $B$ are expressed parametrically using direction angles. Substituting the constraint that $A$ lies on $x + y - 5 = 0$ gives $\frac{1}{\eta_1} = \frac{\cos \theta + \sin \theta}{2}$. Similarly, $B$ on $2x - y - 7 = 0$ gives $\frac{1}{\eta_2} = \frac{2\cos \theta - \sin \theta}{7}$. Using the harmonic mean condition $10 = \frac{2\eta_1 \eta_2}{\eta_1 + \eta_2}$ and solving the resulting trigonometric equation $55\cos \theta + 25\sin \theta = 14$ yields $\cos(\theta - \alpha) = \frac{14}{5\sqrt{146}}$ where $\cos \alpha = \frac{55}{\sqrt{55^2 + 25^2}}$.
Correct Answer: 5