Question

The equations of the assymptotes of the hyperbola $3x^2+10\mathrm{xy}+8y^2+14x+22y+7=0$ are .

• A. $3x^2+10\mathrm{xy}+8y^2+14x+22y-7=0$
• B. $3x^2+10\mathrm{xy}+8y^2+14x+22y=0$
• C. $3x^2+10\mathrm{xy}+8y^2+14x+22y+14=0$
• D. $3x^2+10\mathrm{xy}+8y^2+14x+22y+15=0$

Answer 1

The correct option is D $3x^2+10\mathrm{xy}+8y^2+14x+22y+15=0$

As the equation of the assymptotes and a hyperbola only differs by a constant,
Hence, the equation of the assymptotes will be:
$3x^2+10\mathrm{xy}+8y^2+14x+22y+k=0...\left(i\right)$
k is chosen in such a manner that (i) represents the pair of the straight lines.
Hence, $\mathrm{if}{\mathrm{ax}}^2+2\mathrm{hxy}+{\mathrm{by}}^2+2\mathrm{gx}+2\mathrm{fy}+c=0\mathrm{represents}a\mathrm{pair}\mathrm{of}\mathrm{straight}\mathrm{lines},\mathrm{then}\mathrm{abc}+2\mathrm{fgh}-{\mathrm{af}}^2-{\mathrm{bg}}^2-{\mathrm{ch}}^2=0\Rightarrow 3\times 8\times k+2\times 7\times 11\times 5-3\times 121-8\times 49-k\times 25=0\Rightarrow k=15$
Hence, the combined equation of the assymptotes is :

$3x^2+10\mathrm{xy}+8y^2+14x+22y+15=0$