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 ?
Advertisements
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 slope of the tangent to the curve y = x3 − 3x + 2 at the point whose x-coordinate is 3.
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).
Find the equations of all lines having slope 0 which are tangent to the curve y = `1/(x^2-2x + 3)`
Find points on the curve `x^2/9 + "y"^2/16 = 1` at which the tangent is parallel to x-axis.
Find the equations of the tangent and normal to the given curves at the indicated points:
y = x4 − 6x3 + 13x2 − 10x + 5 at (1, 3)
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 the slope of the tangent and the normal to the following curve at the indicted point y = (sin 2x + cot x + 2)2 at x = π/2 ?
Find the points on the curve y = 3x2 − 9x + 8 at which the tangents are equally inclined with the axes ?
Find the point on the curve y = 3x2 + 4 at which the tangent is perpendicular to the line whose slop is \[- \frac{1}{6}\] ?
At what points on the curve y = x2 − 4x + 5 is the tangent perpendicular to the line 2y + x = 7?
Find the equation of the tangent and the normal to the following curve at the indicated point \[\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1 \text { at } \left( a\sec\theta, b\tan\theta \right)\] ?
Find the equation of the tangent and the normal to the following curve at the indicated point y2 = 4ax at \[\left( \frac{a}{m^2}, \frac{2a}{m} \right)\] ?
Find the equation of the tangent and the normal to the following curve at the indicated point \[\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \text { at } \left( x_1 , y_1 \right)\] ?
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 = a(θ + sinθ), y = a(1 − cosθ) at θ ?
Find the equation of the tangent line to the curve y = x2 − 2x + 7 which perpendicular to the line 5y − 15x = 13. ?
Find the equation of the tangents to the curve 3x2 – y2 = 8, which passes through the point (4/3, 0) ?
Show that the curves 4x = y2 and 4xy = k cut at right angles, if k2 = 512 ?
Find the condition for the following set of curve to intersect orthogonally \[\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \text { and } \frac{x^2}{A^2} - \frac{y^2}{B^2} = 1\] ?
If the straight line xcos \[\alpha\] +y sin \[\alpha\] = p touches the curve \[\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1\] then prove that a2cos2 \[\alpha\] \[-\] b2sin2 \[\alpha\] = p2 ?
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 angle between the curves y = e−x and y = ex at their point of intersections ?
Write the coordinates of the point at which the tangent to the curve y = 2x2 − x + 1 is parallel to the line y = 3x + 9 ?
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 angle between the curves y2 = x and x2 = y at (1, 1) is ______________ .
If the curve ay + x2 = 7 and x3 = y cut orthogonally at (1, 1), then a is equal to _____________ .
The curves y = aex and y = be−x cut orthogonally, if ___________ .
The equation of the normal to the curve x = a cos3 θ, y = a sin3 θ at the point θ = π/4 is __________ .
The angle of intersection of the curves y = 2 sin2 x and y = cos 2 x at \[x = \frac{\pi}{6}\] is ____________ .
Find the angle of intersection of the curves y = 4 – x2 and y = x2.
Show that the line `x/"a" + y/"b"` = 1, touches the curve y = b · e– x/a at the point where the curve intersects the axis of y
`"sin"^"p" theta "cos"^"q" theta` attains a maximum, when `theta` = ____________.
The distance between the point (1, 1) and the tangent to the curve y = e2x + x2 drawn at the point x = 0
The Slope of the normal to the curve `y = 2x^2 + 3 sin x` at `x` = 0 is
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 ______.
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 ______.
