Question

The tangent at $A(2,4)$ on the curve $y=x^3-2x^2+4$ cuts the $x$-axis at $T$, then the length of $AT$ is

• A. $\sqrt{10}$
• B. $\sqrt{12}$
• C. $\sqrt{15}$
• D. $\sqrt{17}$

Answer 1

Answer: (D)

$\sqrt{17}$

Given curve is $y=x^3-2x^2+4$

Therefore, $\frac{dy}{dx}=3x^2-4x$ -------(1)

Given that point A(2,4) passes through the tangent.

Therefore, the slope of the tangent is $3(2{)}^2-4(2)=4$.

Therefore, the equation of tangent is $(y-4)=4(x-2)$

$⟹y-4=4x-8⟹4x-y=4$

Given that the tangent passes through the x-axis.

Therefore, for x-axis $y=0$

Hence, $4x-0=4⟹x=1$

Therefore, point T is $(1,0)$

Hence, length of AT is $\sqrt{(4-0{)}^2+(2-1{)}^2}=\sqrt{16+1}=\sqrt{17}$