Straight Lines
Straight Line
Allen Star Batch
Grade 11
Question:
Two equal sides $OA$ and $OB$ of an isosceles triangle lie in the first quadrant. If the slopes of $OA$ and $OB$ are $\frac{7}{17}$ and $1$, respectively and the length of perpendicular from $O$ to $AB$ is $\sqrt{13}$, if the equation of the side $AB$ is $ax + by = c$ then the value of $(c - a - b)$ is __________.
Step-by-Step Solution
Key Concept: The slope of the perpendicular line can be found using the tangent addition formula applied to the angle between two known slopes.
With slopes of $OA$ and $OB$ as $\frac{7}{17}$ and $1$ respectively, and $OD \perp AB$, we find $\tan\theta = \frac{1 - \frac{7}{17}}{1 + \frac{7}{17}} = \frac{5}{12}$ giving $\tan\frac{\theta}{2} = \frac{1}{5}$. Then $\tan\alpha = \tan(\frac{\theta}{2} + \tan^{-1}\frac{7}{17}) = \frac{2}{3}$, so $\sin\alpha = \frac{2}{\sqrt{13}}$ and $\cos\alpha = \frac{3}{\sqrt{13}}$. The equation of $AB$ is $3x + 2y = 13$.
Correct Answer: 8