Advertisements
Advertisements
Question
Find the points on the curve y = x3 at which the slope of the tangent is equal to the y-coordinate of the point.
Advertisements
Solution
The equation of the given curve is y = x3.
`:. dy/dx = 3x^2`
The slope of the tangent at the point (x, y) is given by,
When the slope of the tangent is equal to the y-coordinate of the point, then y = 3x2.
Also, we have y = x3.
∴3x2 = x3
⇒ x2 (x − 3) = 0
⇒ x = 0, x = 3
When x = 0, then y = 0 and when x = 3, then y = 3(3)2 = 27.
Hence, the required points are (0, 0) and (3, 27).
APPEARS IN
RELATED QUESTIONS
Find the slope of the tangent to curve y = x3 − x + 1 at the point whose x-coordinate is 2.
Find the slope of the tangent to the curve y = x3 − 3x + 2 at the point whose x-coordinate is 3.
Find the equations of all lines having slope 0 which are tangent to the curve y = `1/(x^2-2x + 3)`
Find the equations of the tangent and normal to the given curves at the indicated points:
y = x4 − 6x3 + 13x2 − 10x + 5 at (0, 5)
Find the equation of the normal at the point (am2, am3) for the curve ay2 = x3.
Find the equations of the tangent and normal to the hyperbola `x^2/a^2 - y^2/b^2` at the point `(x_0, y_0)`
Find the slope of the tangent and the normal to the following curve at the indicted point y = x3 − x at x = 2 ?
Find the slope of the tangent and the normal to the following curve at the indicted point x = a (θ − sin θ), y = a(1 − cos θ) at θ = −π/2 ?
Find a point on the curve y = x3 − 3x where the tangent is parallel to the chord joining (1, −2) and (2, 2) ?
Find the points on the curve y2 = 2x3 at which the slope of the tangent is 3 ?
Find the point on the curve y = 3x2 + 4 at which the tangent is perpendicular to the line whose slop is \[- \frac{1}{6}\] ?
Find the points on the curve 2a2y = x3 − 3ax2 where the tangent is parallel to x-axis ?
Find the points on the curve \[\frac{x^2}{9} + \frac{y^2}{16} = 1\] at which the tangent is parallel to y-axis ?
Find the equation of the tangent to the curve \[\sqrt{x} + \sqrt{y} = a\] at the point \[\left( \frac{a^2}{4}, \frac{a^2}{4} \right)\] ?
Find the equation of the tangent and the normal to the following curve at the indicated point\[\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \text{ at }\left( a\cos\theta, b\sin\theta \right)\] ?
Find the equation of the tangent and the normal to the following curve at the indicated point \[\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1 \text { at } \left( x_0 , y_0 \right)\] ?
Find the equation of the tangent and the normal to the following curve at the indicated points \[x = \frac{2 a t^2}{1 + t^2}, y = \frac{2 a t^3}{1 + t^2}\text { at } t = \frac{1}{2}\] ?
Find the equation of the tangent to the curve x2 + 3y − 3 = 0, which is parallel to the line y= 4x − 5 ?
Find the angle of intersection of the following curve y2 = x and x2 = y ?
Find the angle of intersection of the following curve x2 + y2 − 4x − 1 = 0 and x2 + y2 − 2y − 9 = 0 ?
Find the angle of intersection of the following curve x2 = 27y and y2 = 8x ?
Find the condition for the following set of curve to intersect orthogonally \[\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \text { and } \frac{x^2}{A^2} - \frac{y^2}{B^2} = 1\] ?
Find the point on the curve y = x2 − 2x + 3, where the tangent is parallel to x-axis ?
The slope of the tangent to the curve x = 3t2 + 1, y = t3 −1 at x = 1 is ___________ .
Any tangent to the curve y = 2x7 + 3x + 5 __________________ .
Find the equation of tangent to the curve y = x2 +4x + 1 at (-1 , -2).
Find the equation of tangent to the curve `y = sqrt(3x -2)` which is parallel to the line 4x − 2y + 5 = 0. Also, write the equation of normal to the curve at the point of contact.
Show that the equation of normal at any point on the curve x = 3cos θ – cos3θ, y = 3sinθ – sin3θ is 4 (y cos3θ – x sin3θ) = 3 sin 4θ
The point on the curve y2 = x, where the tangent makes an angle of `pi/4` with x-axis is ______.
If the straight line x cosα + y sinα = p touches the curve `x^2/"a"^2 + y^2/"b"^2` = 1, then prove that a2 cos2α + b2 sin2α = p2.
The equation of normal to the curve 3x2 – y2 = 8 which is parallel to the line x + 3y = 8 is ______.
If the curve ay + x2 = 7 and x3 = y, cut orthogonally at (1, 1), then the value of a is ______.
The tangent to the curve y = e2x at the point (0, 1) meets x-axis at ______.
The points on the curve `"x"^2/9 + "y"^2/16` = 1 at which the tangents are parallel to the y-axis are:
The two curves x3 - 3xy2 + 5 = 0 and 3x2y - y3 - 7 = 0
Tangent is drawn to the ellipse `x^2/27 + y^2 = 1` at the point `(3sqrt(3) cos theta, sin theta), 0 < 0 < 1`. The sum of the intercepts on the axes made by the tangent is minimum if 0 is equal to
The Slope of the normal to the curve `y = 2x^2 + 3 sin x` at `x` = 0 is
If the tangent to the conic, y – 6 = x2 at (2, 10) touches the circle, x2 + y2 + 8x – 2y = k (for some fixed k) at a point (α, β); then (α, β) is ______.
