Question

If slope of tangent to curve y = x³ at a point is equal to ordinate of point, then the point is _________________.

Answer 1

The equation of given curve is y = x³.

Let (h, k) be the point on the curve at which slope of tangent is equal to ordinate of the point.

∴ k = h³ .....(1)

Now,

y = x³

Differentiating both sides with respect to x, we get

$\frac{dy}{dx}=3x^2$

∴ Slope of tangent at (h, k) = ${\left(\frac{dy}{dx}\right)}_{\left(h,k\right)}=3h^2$

It is given that,

Slope of tangent at (h, k) = Ordinate of (h, k)

⇒ 3h² = k .....(2)

From (1) and (2), we get

$3h^2=h^3$

$\Rightarrow h^3-3h^2=0$

$\Rightarrow h^2\left(h-3\right)=0$

⇒ h = 0 or h − 3 = 0

⇒ h = 0 or h = 3

When h = 0,

k = (0)³ = 0 [From (1)]

When h = 3,

k = (3)³ = 27 [From (1)]

So, the coordinates of the required points are (0, 0) and (3, 27).

Thus, if slope of tangent to curve y = x³ at a point is equal to ordinate of point, then the points are (0, 0) and (3, 27).


If slope of tangent to curve y = x³ at a point is equal to ordinate of point, then the point is ___(0, 0) and (3, 27)___.