Question

The equation $3(x-1{)}^2+2h(x-1\left)\right(y-2)+3(y-2{)}^2=0$ represents a pair of straight lines passing through the point $(1,2)$. The two lines are real and distinct if $h^2$

• A. is greater than $3$
• B. is greater than $9$
• C. equals $7$
• D. is greater than $7$

Answer 1

Answer: (B)

is greater than $9$

$3(x-1{)}^2+2h(x-1\left)\right(y-2)+3(y-2{)}^2=0$

for shifting of origin

det $(x-1)=x,(y-2)=y$

$\therefore 3x^2+2hxy+3y^2=0$ - ... equation (1)

is an equation of pair of straight lines

passing through origin i.e $x=0$ and $y=0$

$x=0$

$\Rightarrow x-1=0\Rightarrow x=1$

$y=0$

$\Rightarrow y-2=0\Rightarrow y=2$

so given pair of straight lines passes through

the point (1,2) .

Comparing equation (1) with general form of

pair of straight lines passing through origin

i.e $a{x}^2+2hxy+b{y}^2=0$

$\therefore a=3b=3$

to find lines in given equation

$y=\left(\frac{-h\pm \sqrt{h^2-ab}}{b}\right)x$

for both lines to be real and distinct

$\Rightarrow h^2-ab>0$

$\Rightarrow h^2-9>0$

$\Rightarrow h^2>9$

So. Answer: option (B)