Question

If the equation $2{{x}^{2}}-2hxy+2{{y}^{2}}=0$ represents two coincident straight lines then the value of h is equal to
A. $\pm 6$
B. $6-\sqrt{6}$
C. $-6-\sqrt{6}$
D. $\pm 2$

Answer 1

Hint: The nature of the straight line can be determined from the value of ${{h}^{2}}-ab$ . The value of ${{h}^{2}}-ab$ is zero for two coincident straight lines. 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 $a{{x}^{2}}+2hxy+b{{y}^{2}}$ .





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




Complete step by step solution:The given equation of the straight line is $2{{x}^{2}}-2hxy+2{{y}^{2}}=0$.
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-
$a=2,h=-h,b=2$
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 coincident.
The value of ${{h}^{2}}-ab$ is equal to zero for real and coincident straight lines. Thus we can write-
$
   {{h}^{2}}-ab=0 \\
  {{(-h)}^{2}}-2\times 2=0 \\
  {{h}^{2}}-4=0 \\
  {{h}^{2}}=4 \\
  h=\pm \sqrt{4} \\
  h=\pm 2 \\
$
So, the value of $h$ is $\pm 2$.
Thus we can write that if the equation $2{{x}^{2}}-2hxy+2{{y}^{2}}=0$ represents two coincident straight lines then the value of h is equal to $\pm 2$ .


Option ‘D’ is correct



Note:Two straight lines are real and coincident means that the straight lines meets at a common point. The other conditions and nature of a straight line are if the value of ${{h}^{2}}-ab>0$ then the equation is real and distinct and if the value of ${{h}^{2}}-ab<0$ then the equation is imaginary.