Question

Show that the following equation represents a pair of lines.find. Find also the angle between them 6x2+13xy+6y2+8x+7y+2=0

Answer 1

No the equations don't represent a pair of lines....

The angle in that given case not findable as there is only one equation...

Answer 2

The angle between the given pair of straight lines = 22.62°

Step-by-step explanation:

The given equation of line = $6x^{2} + 13xy + 6y^{2} + 8x + 7y + 2 = 0$

General form of the equation is $ax^{2} + 2hxy + by^{2} + 2gx + 2fy + c = 0$

Compare both equations we get,

a = 6, b = 6, h = $\frac{13}{2}$ , g = 4, $f = \frac{7}{2}$ , c = 2

The condition for the given equation represents a pair of lines is

$abc + 2fgh - af^{2} - bg^{2} - ch^{2} = 0$ -------- (1)

Put the values of constants in above equation we get,

⇒ 6 × 6 × 2 - 2 × $\frac{7}{2}$  × 4 × $\frac{13}{2}$ - 6 × $[\frac{7}{2}] ^{2}$ - 6 × 16 - 2 × $[\frac{13}{2} ]^{2}$ = 0

⇒ Thus the condition given in equation (1) is satisfied. so this equation represents the pair of lines.

Now the angle between them is calculated by the following formula,

$\tan \theta =\frac{2 \sqrt{(h^{2} - ab )} }{a + b}$

⇒ Put all the values in above formula we get,

⇒  $\tan \theta =\frac{2 \sqrt{(\frac{13}{2} ^{2} - 36 )} }{6 + 6}$

⇒   $\tan \theta = \frac{5}{12}$

⇒  $\theta = 22.62$ °

This is the angle between the given pair of straight lines.