Advertisements
Advertisements
प्रश्न
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 ?
उत्तर
Slope of x - axis is 0
Let (x1, y1) be the required point.
\[y = 2 x^3 - 15 x^2 + 36x - 21\]
\[\text { Since }\left( x_1 , y_1 \right) \text { lies on the curve . Therefore } \]
\[ y_1 = 2 {x_1}^3 - 15 {x_1}^2 + 36 x_1 - 21 . . . \left( 1 \right)\]
\[\text { Now,} y = 2 x^3 - 15 x^2 + 36x - 21\]
\[ \Rightarrow \frac{dy}{dx} = 6 x^2 - 30x + 36\]
\[\text { Slope of tangent at }\left( x_1 , y_1 \right)= \left( \frac{dy}{dx} \right)_\left( x_1 , y_1 \right) = 6 {x_1}^2 - 30 x_1 + 36\]
\[\text { Given that }\]
\[\text { Slope of tangent at }\left( x, y \right)= \text { slope of thex-axis }\]
\[6 {x_1}^2 - 30 x_1 + 36 = 0\]
\[ \Rightarrow {x_1}^2 - 5 x_1 + 6 = 0\]
\[ \Rightarrow \left( x_1 - 2 \right)\left( x_1 - 3 \right) = 0\]
\[ \Rightarrow x_1 = 2 \text{ or }x_1 = 3\]
\[\text { Case }1: x_1 = 2\]
\[ y_1 = 16 - 60 + 72 - 21 = 7 ...............[\text { From } (1)]\]
\[\left( x_1 , y_1 \right) = \left( 2, 7 \right)\]
\[\text { Equation of tangent is },\]
\[y - y_1 = m\left( x - x_1 \right)\]
\[ \Rightarrow y - 7 = 0\left( x - 2 \right)\]
\[ \Rightarrow y = 7\]
\[\text { Case }2: x_1 = 3\]
\[ y_1 = 54 - 135 + 108 - 21 = 6 .................[\text { From }(1)]\]
\[\left( x_1 , y_1 \right) = \left( 3, 6 \right)\]
\[\text { Equation of tangent is },\]
\[y - y_1 = m\left( x - x_1 \right)\]
\[ \Rightarrow y - 6 = 0\left( x - 3 \right)\]
\[ \Rightarrow y = 6\]
APPEARS IN
संबंधित प्रश्न
Find the equations of the tangent and normal to the curve `x^2/a^2−y^2/b^2=1` at the point `(sqrt2a,b)` .
Find the slope of the tangent to curve y = x3 − x + 1 at the point whose x-coordinate is 2.
Find points at which the tangent to the curve y = x3 − 3x2 − 9x + 7 is parallel to the x-axis.
Find the point on the curve y = x3 − 11x + 5 at which the tangent is y = x − 11.
Find the points on the curve y = x3 at which the slope of the tangent is equal to the y-coordinate of the point.
The slope of the normal to the curve y = 2x2 + 3 sin x at x = 0 is
(A) 3
(B) 1/3
(C) −3
(D) `-1/3`
Find the slope of the tangent and the normal to the following curve at the indicted point y = 2x2 + 3 sin x at x = 0 ?
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 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 y = x4 − 6x3 + 13x2 − 10x + 5 at x = 1?
Find the equation of the tangent and the normal to the following curve at the indicated point \[\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \text { at } \left( x_1 , y_1 \right)\] ?
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 line to the curve y = x2 + 4x − 16 which is parallel to the line 3x − y + 1 = 0 ?
Find the equations of all lines of slope zero and that are tangent to the curve \[y = \frac{1}{x^2 - 2x + 3}\] ?
Prove that \[\left( \frac{x}{a} \right)^n + \left( \frac{y}{b} \right)^n = 2\] touches the straight line \[\frac{x}{a} + \frac{y}{b} = 2\] for all n ∈ N, at the point (a, b) ?
Find the angle of intersection of the following curve \[\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1\] and x2 + y2 = ab ?
Find the angle of intersection of the following curve x2 + y2 = 2x and y2 = x ?
Show that the following set of curve intersect orthogonally x3 − 3xy2 = −2 and 3x2y − y3 = 2 ?
Prove that the curves xy = 4 and x2 + y2 = 8 touch each other ?
If the tangent to a curve at a point (x, y) is equally inclined to the coordinates axes then write the value of \[\frac{dy}{dx}\] ?
Write the equation of the tangent drawn to the curve \[y = \sin x\] at the point (0,0) ?
The equations of tangent at those points where the curve y = x2 − 3x + 2 meets x-axis are _______________ .
The angle of intersection of the curves xy = a2 and x2 − y2 = 2a2 is ______________ .
The curves y = aex and y = be−x cut orthogonally, if ___________ .
If the curves y = 2 ex and y = ae−x intersect orthogonally, then a = _____________ .
The angle of intersection of the parabolas y2 = 4 ax and x2 = 4ay at the origin is ____________ .
The normal to the curve x2 = 4y passing through (1, 2) is _____________ .
Find the equation of a tangent and the normal to the curve `"y" = (("x" - 7))/(("x"-2)("x"-3)` at the point where it cuts the x-axis
Find the angle of intersection of the curves y2 = 4ax and x2 = 4by.
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.
At what points on the curve x2 + y2 – 2x – 4y + 1 = 0, the tangents are parallel to the y-axis?
`"sin"^"p" theta "cos"^"q" theta` attains a maximum, when `theta` = ____________.
The number of values of c such that the straight line 3x + 4y = c touches the curve `x^4/2` = x + y is ______.
Two vertical poles of heights, 20 m and 80 m stand apart on a horizontal plane. The height (in meters) of the point of intersection of the lines joining the top of each pole to the foot of the other, From this horizontal plane is ______.
The curve `(x/a)^n + (y/b)^n` = 2, touches the line `x/a + y/b` = 2 at the point (a, b) for n is equal to ______.
The normal of the curve given by the equation x = a(sinθ + cosθ), y = a(sinθ – cosθ) at the point θ is ______.
Find the equation to the tangent at (0, 0) on the curve y = 4x2 – 2x3
