Question

The nature of straight lines represented by the equation \[4{{x}^{2}}+12xy+9{{y}^{2}}=0\] , is
A. Real and coincident
B. Real and different
C. Imaginary and different
D. None of the above

Answer 1

Hint: The nature of the straight line can be determined from the value of ${{h}^{2}}-ab$ . Here we have to determine the value of ${{h}^{2}}-ab$ by comparing the given equation of straight line with the general formula of straight line that is $a{{x}^{2}}+2hxy+b{{y}^{2}}$ we can get the values of $h,a,b$ .



Formula Used:$a{{x}^{2}}+2hxy+b{{y}^{2}}=0$



Complete step by step solution:The given equation of the straight line is \[4{{x}^{2}}+12xy+9{{y}^{2}}=0\].
The general equation of a pair of straight line is$a{{x}^{2}}+2hxy+b{{y}^{2}}=0$.
Comparing the given equation with the general equation we will get the following values as follows-
$a=4,h=6,b=9$
Now the nature of a straight line can be determined by the ${{h}^{2}}-ab$ value. The following conditions are as follows-
If ${{h}^{2}}-ab>0$ then the equation is real and distinct.
If ${{h}^{2}}-ab=0$ then the equation is real and coincident.
If ${{h}^{2}}-ab<0$ then the equation is imaginary.
The value of ${{h}^{2}}-ab$ is equal to
   ${{h}^{2}}-ab \\ $
 $={{6}^{2}}-9\times 4 \\$
 $ =36-36 \\ $
 $ =0 \\ $

So, the value of ${{h}^{2}}-ab$ is zero. Thus the straight lines are real and coincident.
Thus we can write that the nature of straight lines represented by the equation \[4{{x}^{2}}+12xy+9{{y}^{2}}=0\] , is real and coincident.



Option ‘A’ is correct



Note: Two straight lines are real and coincident means that the lines meet at a point. Thus we will get a intercept point.