First slide

Equation of line in Straight lines

Question

Consider three lines as follows.
$L_{1} : 5 x - y + 4 = 0$
$L_{2} : 3 x - y + 5 = 0$
$L_{3} : x + y + 8 = 0$
If these lines enclose a triangle ABC and sum of the square of the tangent to the angles can be expressed in the form p/q, where p and Q are relatively prime numbers, then the value of p+q is

Moderate

A

500
B

450
C

230
D

465

Solution

Arranging the lines in descending order of slope, we have
$m_{1} = 5, m_{2} = 3, and m_{3} = - 1$
$∴ tanA = \frac{m_{1} - m_{2}}{1 + m_{1} m_{2}} = \frac{2}{1 + 15} = \frac{1}{8}$
$tanB = \frac{m_{2} - m_{3}}{1 + m_{2} m_{3}} = \frac{3 + 1}{1 - 3} = - 2$
$tanC = \frac{m_{3} - m_{1}}{1 + m_{3} m_{1}} = \frac{- 1 - 5}{1 - 5} = \frac{3}{2}$
$\sum \tan^{2} A = \frac{1}{64} + 4 + \frac{9}{4} = \frac{1 + 256 + 144}{64} = \frac{401}{64}$
or $p + q = 465$

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