# The point of lines represented by  $\dpi{100} 3ax^{2}+5xy+(a^{2}-2)y^{2}=0$   and $\dpi{100} \perp$  to each other for Option 1) two values of $a$    Option 2) $\forall \; \; a$ Option 3) for one value of $a$ Option 4) for no values of $a$

P Plabita

As we learnt in

General equation of a conic -

$ax^{2}+2hxy+by^{2}+2gx+2fy+c= 0$

- wherein

$a,b,c, f,g,h$  are constants

$3ax^{2}+5xy+\left ( a^{2}-2 \right )y^{2}=0$

For perpendicular lines, sum of coefficients of x and y=0

$\\ a^{2}-2+3a=0 \\ \\ a^{2}+3a-2=0$

So, there are two real values of a.

Option 1)

two values of $a$

This option is correct

Option 2)

$\forall \; \; a$

This option is incorrect

Option 3)

for one value of $a$

This option is incorrect

Option 4)

for no values of $a$

This option is incorrect

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