Question

Distance between the parallel lines $(x+2y{)}^2+13\sqrt{5}(x+2y)+180=0$ is

• A. $\sqrt{5}$
• B. $4$$\sqrt{5}$
• C. $5$
• D. $4$

Answer 1

The correct option is C $5$

Given $(x+2y{)}^2+13\sqrt{5}(x+2y)+180=0$

$⟹x^2+4y^2+4xy+13\sqrt{5}x+26\sqrt{5}y+180=0$

It is of the form $a{x}^2+2hxy+b{y}^2+2gx+2fy+c=0$

On comparing, we get

$a=1,h=4,b=4,g=\frac{13\sqrt{5}}{2},f=13\sqrt{5},c=180$

Now, for lines to be parallel

$h=ab$ and $a{f}^2=b{g}^2$

Since,these two equations are satisfied for the above values of $a,h,b,g,f,c$

Hence, the two lines are parallel.

Now applying the formula to calculate the distance between parallel pair of lines.

Distance $=2\sqrt{\frac{g^2-ac}{a(a+b)}}$

$=2\sqrt{\frac{(\frac{13\sqrt{5}}{2}{)}^2-(1\left)\right(180)}{1(1+4)}}$

$⟹$ distance= $5$.

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