Advertisements
Advertisements
प्रश्न
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
उत्तर
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
संबंधित प्रश्न
Show that the equation of normal at any point t on the curve x = 3 cos t – cos3t and y = 3 sin t – sin3t is 4 (y cos3t – sin3t) = 3 sin 4t
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 points at which the tangent to the curve y = x3 − 3x2 − 9x + 7 is parallel to the x-axis.
Find the equation of all lines having slope 2 which are tangents to the curve `y = 1/(x- 3), x != 3`
Find the equation of the tangent line to the curve y = x2 − 2x + 7 which is perpendicular to the line 5y − 15x = 13.
Find the equation of the normal at the point (am2, am3) for the curve ay2 = x3.
Find the equation of the normal to curve y2 = 4x at the point (1, 2).
Find the slope of the tangent and the normal to the following curve at the indicted point \[y = \sqrt{x} \text { at }x = 9\] ?
Find the points on the curve y2 = 2x3 at which the slope of the tangent is 3 ?
Find the points on the curve \[\frac{x^2}{4} + \frac{y^2}{25} = 1\] at which the tangent is parallel to the x-axis ?
Find the points on the curve y = x3 where the slope of the tangent is equal to the x-coordinate of the point ?
Find the equation of the tangent and the normal to the following curve at the indicated point \[c^2 \left( x^2 + y^2 \right) = x^2 y^2 \text { at }\left( \frac{c}{\cos\theta}, \frac{c}{\sin\theta} \right)\] ?
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 point 4x2 + 9y2 = 36 at (3cosθ, 2sinθ) ?
Find the equation of the tangent and the normal to the following curve at the indicated points:
x = 3cosθ − cos3θ, y = 3sinθ − sin3θ?
Determine the equation(s) of tangent (s) line to the curve y = 4x3 − 3x + 5 which are perpendicular to the line 9y + x + 3 = 0 ?
Find the equations of all lines of slope zero and that are tangent to the curve \[y = \frac{1}{x^2 - 2x + 3}\] ?
At what points will be tangents to the curve y = 2x3 − 15x2 + 36x − 21 be parallel to x-axis ? Also, find the equations of the tangents to the curve at these points ?
Find the angle of intersection of the following curve y = x2 and x2 + y2 = 20 ?
Show that the following set of curve intersect orthogonally x3 − 3xy2 = −2 and 3x2y − y3 = 2 ?
Show that the curves 2x = y2 and 2xy = k cut at right angles, if k2 = 8 ?
Find the condition for the following set of curve to intersect orthogonally \[\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \text { and } \frac{x^2}{A^2} - \frac{y^2}{B^2} = 1\] ?
Write the slope of the normal to the curve \[y = \frac{1}{x}\] at the point \[\left( 3, \frac{1}{3} \right)\] ?
The equation of the normal to the curve y = x(2 − x) at the point (2, 0) is ________________ .
The point on the curve y2 = x where tangent makes 45° angle with x-axis is ____________________ .
The point at the curve y = 12x − x2 where the slope of the tangent is zero will be _____________ .
The curves y = aex and y = be−x cut orthogonally, if ___________ .
The point on the curve y = 6x − x2 at which the tangent to the curve is inclined at π/4 to the line x + y= 0 is __________ .
Find the angle of intersection of the curves \[y^2 = 4ax \text { and } x^2 = 4by\] .
The two curves x3 – 3xy2 + 2 = 0 and 3x2y – y3 = 2 ______.
The tangent to the curve given by x = et . cost, y = et . sint at t = `pi/4` makes with x-axis an angle ______.
Find an angle θ, 0 < θ < `pi/2`, which increases twice as fast as its sine.
Prove that the curves xy = 4 and x2 + y2 = 8 touch each other.
The points at which the tangents to the curve y = x3 – 12x + 18 are parallel to x-axis are ______.
The tangent to the parabola x2 = 2y at the point (1, `1/2`) makes with the x-axis an angle of ____________.
Find a point on the curve y = (x – 2)2. at which the tangent is parallel to the chord joining the points (2, 0) and (4, 4).
The slope of the tangentto the curve `x= t^2 + 3t - 8, y = 2t^2 - 2t - 5` at the point `(2, -1)` is
The number of values of c such that the straight line 3x + 4y = c touches the curve `x^4/2` = x + y is ______.
