NIMCET 2018 — Mathematics PYQ

Concept:
- For a pair of lines ax² + 2hxy + by² = 0:
  Sum of the slopes is: $-\frac{2h}{b}$.
  Product of the slopes is: $\frac{a}{b}$.
- (a - b)² = (a + b)² - 4ab.

Calculation:

The given equation of the pair of lines is 6x${}^2-2$xy- 2$y^2$= 0. 
Comparing this with the general equation of a pair of lines ax${}^2+2$hxy+by${}^2=0$, we have:
a= 6, 2h= - 1 and b= - 2. 
Let the slopes of the lines be p and q.
Product of slopes is: \frac ab.

$\Rightarrow$pq$=\frac{6}{-2}=-3$ ...(1)

$Sumofslopesis:-\frac{2\mathrm{h}}{\mathrm{b}}.$
$$\Rightarrow \mathrm{p}+\mathrm{q}=-\frac{-1}{-2}=-\frac{1}{2}$$
...(2)
Now, ( p- q${)}^2$= ( p+ q${)}^2$- 4pq. 
Using equations (1) and (2), we get:
$$\begin{array}{rl}& amp;\text{esing aqations (1) ano (2), mo g}\\ & amp;\Rightarrow (\mathrm{p}-\mathrm{q}{)}^2={\left(-\frac{1}{2}\right)}^2-4(-3)\\ & amp;\Rightarrow (\mathrm{p}-\mathrm{q}{)}^2=\frac{1}{4}+12=\frac{49}{4}\\ & amp;\Rightarrow \mathrm{p}-\mathrm{q}=\pm \frac{7}{2}\\ & amp;\cdot \text{The difference betryeen the slon}\end{array}$$
$\therefore$The difference between the slopes of both the lines is $\frac{7}{2}.$