Question

The point(s) at each of which the tangents to the curve $y=x^3-3x^2-7x+6$ cut off on the positive semi axis OX a line segment half that on the negative semi axis OY then the co-ordinates the point(s) is/ are given by:

• A. $(-1,9)$
• B. $(3,-15)$
• C. $(1,-3)$
• D. none

Answer 1

Answer: (B)

If $OA=a;OB=2a\Rightarrow \tan \theta =2$
slope of the tangent is 2
$\Rightarrow {\left(\frac{dy}{dx}\right)}_{x_1{y}_1}=2\Rightarrow 3x_1^2-6x_1-7=2\Rightarrow 3x_1^2-2x_1-3=0$
$\Rightarrow x_1=3\text{ or }-1\text{ (rejected) }$
$\Rightarrow (3,-15)\Rightarrow B]$
[Also for the point $P;x\&y$ both positive or $x$ positive \& $y$ negative or $x\&y$ both negative ]