Question

Assertion :The equation $3x^2-2y^2+4x-6y=0$ represents a hyperbola. Reason: The second degree equation $a{x}^2+2hxy+b{y}^2+2gx+2fy+c=0$ represents a hyperbola if $abc+2fgh-a{f}^2-b{g}^2-c{h}^2\ne 0$ & $h^2>ab$.

• A. Both Assertion and Reason are correct and Reason is the correct explanation for Assertion
• B. Both Assertion and Reason are correct but Reason is not the correct explanation for Assertion
• C. Assertion is correct but Reason is incorrect
• D. Both Assertion and Reason are incorrect

Answer 1

The correct option is A Both Assertion and Reason are correct and Reason is the correct explanation for Assertion

Given equation of curve is $3x^2-2y^2+4x-6y=0$
$(a=3,b=-2,h=0=c,g=2,f=-3)$
$\therefore \Delta =abc+2fgh-a{f}^2-b{g}^2-c{h}^2$ $=0+0-3(-3{)}^2-(-2\left)\right(2{)}^2-0=-27+8$ $=-19\ne 0$
&$h^2=0,ab=-6$ So $h^2>ab$ as
$0>-6$
Hence, given curve represents a Hyperbola.
Clearly Assertion & Reason both are true and Reason is correct explanation of Assertion.