Question

The graphs $y-2=2x(x^2-2)$ and $y+1=x(x^2+2)$ intersect at exactly three distinct points. If all the three points are collinear, then the slope of the line joining these points is

• A. $2$
• B. $4$
• C. $6$
• D. $8$

Answer 1

The correct option is D $8$

$y-2=2x(x^2-2)$ and $y+1=x(x^2+2)$
Let points of intersection be $P(x_1,y_1),Q(x_2,y_2)$ and $R(x_3,y_3)$
So, $y_1=2x_1^3-4x_1+2\cdots \left(1\right)$
and $y_1=x_1^3+2x_1-1\cdots \left(2\right)$
From equations $\left(1\right)$ and $\left(2\right),$ we get
$-y_1=-8x_1+4\Rightarrow y_1=8x_1-4$
Similarly, $y_2=8x_2-4$

Now, $y_2-y_1=8(x_2-x_1)$
$\therefore \frac{y_2-y_1}{x_2-x_1}=8$
Hence, the slope of the required line is $8.$