Question

STATEMENT-1 :There lies exactly $3$ unique points on the curve $8x^3+y^3+6xy=1$ which form an equilateral triangle.
STATEMENT-2 : The curve $8x^3+y^3+6xy=1$ consists of a straight line and a point which does not lie on the line.

• A. STATEMENT -1 is True, STATEMENT -2 is True; STATEMENT-2 is a correct explanation for STATEMENT - 1
• B. STATEMENT -1 is True,STATEMENT -2 is True; STATEMENT -2 is NOT a correct explanation for STATEMENT - 1
• C. STATEMENT-1 is True, STATEMENT-2 is False
• D. STATEMENT-1 is False, STATEMENT-2 is True

Answer 1

Answer: (A)

STATEMENT -1 is True, STATEMENT -2 is True; STATEMENT-2 is a correct explanation for STATEMENT - 1

We have,
$y^3+8x^3=1-6xy$

adding both the sides $6x{y}^2+12x^{2}y$,
$y^3+6x{y}^2+12x^{2}y+8x^3=1-6xy+6x{y}^2+12x^{2}y$
$(y+2x{)}^3=1-6xy+6x{y}^2+12x^{2}y$
$(y+2x{)}^3-1^3=6xy(-1+y+2x)$
$(y+2x-1)\left(\right(y+2x{)}^2+(y+2x)+1)=6xy(2x+y-1)$
$(y+2x-1)(y^2+4x^2+4xy+y+2x+1)=6xy(2x+y-1)$
$(y+2x-1)(y^2+4x^2-2xy+y+2x+1)=0$
$(y^2+4x^2-2xy+y+2x+1)=0$
$y^2+y(1-2x)+4x^2+2x+1=0$
$D={(1-2x)}^2-4(4x^2+2x+1)=-3{(2x+1)}^2$
For real $y$
$D=0$
and $x=-\frac{1}{2}$ and $y=1$.
So there is only one point which doesn't lie on the straight line.
Hence Assertion and Reason both are correct and reason is correct explanation.