Advertisements
Advertisements
प्रश्न
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?
उत्तर
\[y= x^4 - 6 x^3 + 13 x^2 - 10x + 5\]
\[\text{ When }x = 1 , \]
`y = 1 - 6 + 13 - 10 + 5 = 3`
\[\text { So}, \left( x_1 , y_1 \right) = \left( 1, 3 \right)\]
\[\text { Now,} y= x^4 - 6 x^3 + 13 x^2 - 10x + 5\]
\[\text { Differentiating both sides w.r.t.x,} \]
\[\frac{dy}{dx} = 4 x^3 - 18 x^2 + 26x - 10\]
\[\text { Slope of tangent },m= \left( \frac{dy}{dx} \right)_\left( 1, 3 \right) =4-18+26 - 10 = 2\]
\[\text { Equation of tangent is },\]
\[y - y_1 = 2 \left( x - x_1 \right)\]
\[ \Rightarrow y - 3 = 2\left( x - 1 \right)\]
\[ \Rightarrow y - 3 = 2x - 2\]
\[ \Rightarrow 2x - y + 1 = 0\]
\[\text { Equation of normal is
},\]
\[y - y_1 = \frac{- 1}{m} \left( x - x_1 \right)\]
\[ \Rightarrow y - 3 = \frac{- 1}{2} \left( x - 1 \right)\]
\[ \Rightarrow 2y - 6 = - x + 1\]
\[ \Rightarrow x + 2y - 7 = 0\]
APPEARS IN
संबंधित प्रश्न
Find the slope of the tangent to curve y = x3 − x + 1 at the point whose x-coordinate is 2.
Find the point on the curve y = x3 − 11x + 5 at which the tangent is y = x − 11.
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 points on the curve y = x3 at which the slope of the tangent is equal to the y-coordinate of the point.
For the curve y = 4x3 − 2x5, find all the points at which the tangents passes through the origin.
Find the slope of the tangent and the normal to the following curve at the indicted point \[y = \sqrt{x^3} \text { at } x = 4\] ?
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 point on the curve y = x2 where the slope of the tangent is equal to the x-coordinate of the point ?
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 \[y^2 = \frac{x^3}{4 - x}at \left( 2, - 2 \right)\] ?
Find the equation of the tangent and the normal to the following curve at the indicated point x2 = 4y at (2, 1) ?
Find the equation of the tangent and the normal to the following curve at the indicated point y2 = 4ax at (x1, y1)?
Find the equation of the tangent and the normal to the following curve at the indicated points x = θ + sinθ, y = 1 + cosθ at θ = \[\frac{\pi}{2}\] ?
Find the equation of the tangent and the normal to the following curve at the indicated points:
x = 3cosθ − cos3θ, y = 3sinθ − sin3θ?
The equation of the tangent at (2, 3) on the curve y2 = ax3 + b is y = 4x − 5. Find the values of a and b ?
Find the equation of the tangent to the curve x = sin 3t, y = cos 2t at
\[t = \frac{\pi}{4}\] ?
Find the angle of intersection of the following curve x2 = 27y and y2 = 8x ?
If the tangent to a curve at a point (x, y) is equally inclined to the coordinates axes then write the value of \[\frac{dy}{dx}\] ?
Find the coordinates of the point on the curve y2 = 3 − 4x where tangent is parallel to the line 2x + y− 2 = 0 ?
Write the angle between the curves y2 = 4x and x2 = 2y − 3 at the point (1, 2) ?
Write the slope of the normal to the curve \[y = \frac{1}{x}\] at the point \[\left( 3, \frac{1}{3} \right)\] ?
The point at the curve y = 12x − x2 where the slope of the tangent is zero will be _____________ .
The equations of tangent at those points where the curve y = x2 − 3x + 2 meets x-axis are _______________ .
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
Find an angle θ, 0 < θ < `pi/2`, which increases twice as fast as its sine.
The two curves x3 – 3xy2 + 2 = 0 and 3x2y – y3 – 2 = 0 intersect at an angle of ______.
The slope of the tangent to the curve x = a sin t, y = a{cot t + log(tan `"t"/2`)} at the point ‘t’ is ____________.
The tangent to the curve y = x2 + 3x will pass through the point (0, -9) if it is drawn at the point ____________.
Find the equation of the tangent line to the curve y = x2 − 2x + 7 which is parallel to the line 2x − y + 9 = 0.
If `tan^-1x + tan^-1y + tan^-1z = pi/2`, then
The number of common tangents to the circles x2 + y2 – 4x – 6x – 12 = 0 and x2 + y2 + 6x + 18y + 26 = 0 is
The number of values of c such that the straight line 3x + 4y = c touches the curve `x^4/2` = x + y is ______.
The normal of the curve given by the equation x = a(sinθ + cosθ), y = a(sinθ – cosθ) at the point θ is ______.
