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?
Advertisements
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 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.
Show that the tangents to the curve y = 7x3 + 11 at the points where x = 2 and x = −2 are parallel.
Find the points on the curve y = x3 at which the slope of the tangent is equal to the y-coordinate of the point.
Find the equation of the normal at the point (am2, am3) for the curve ay2 = x3.
Prove that the curves x = y2 and xy = k cut at right angles if 8k2 = 1. [Hint: Two curves intersect at right angle if the tangents to the curves at the point of intersection are perpendicular to each other.]
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 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 x2 + 3y + y2 = 5 at (1, 1) ?
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 y = x3 − 2x2 − 2x at which the tangent lines are parallel to the line y = 2x− 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 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 y2 = 4ax at (x1, y1)?
Find the equation of the tangent to the curve x = θ + sin θ, y = 1 + cos θ at θ = π/4 ?
Find the equation of the tangent and the normal to the following curve at the indicated points:
x = 3cosθ − cos3θ, y = 3sinθ − sin3θ?
Find the equation of the normal to the curve ay2 = x3 at the point (am2, am3) ?
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 equation of the tangent to the curve x2 + 3y − 3 = 0, which is parallel to the line y= 4x − 5 ?
Show that the curves 2x = y2 and 2xy = k cut at right angles, if k2 = 8 ?
Prove that the curves xy = 4 and x2 + y2 = 8 touch each other ?
If the tangent line at a point (x, y) on the curve y = f(x) is parallel to y-axis, find the value of \[\frac{dx}{dy}\] ?
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 y = e−x and y = ex at their point of intersections ?
The slope of the tangent to the curve x = t2 + 3 t − 8, y = 2t2 − 2t − 5 at point (2, −1) is ________________ .
The normal to the curve x2 = 4y passing through (1, 2) is _____________ .
Find the equation of tangents to the curve y = cos(x + y), –2π ≤ x ≤ 2π that are parallel to the line x + 2y = 0.
Find the equation of tangent to the curve y = x2 +4x + 1 at (-1 , -2).
Find the angle of intersection of the curves y2 = x and x2 = y.
The two curves x3 – 3xy2 + 2 = 0 and 3x2y – y3 – 2 = 0 intersect at an angle of ______.
The two curves x3 - 3xy2 + 5 = 0 and 3x2y - y3 - 7 = 0
The distance between the point (1, 1) and the tangent to the curve y = e2x + x2 drawn at the point x = 0
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 line y = x + 1 is a tangent to the curve y2 = 4x at the point
If `tan^-1x + tan^-1y + tan^-1z = pi/2`, then
Tangent and normal are drawn at P(16, 16) on the parabola y2 = 16x, which intersect the axis of the parabola at A and B, respectively. If C is the centre of the circle through the points P, A and B and ∠CPB = θ, then a value of tan θ is:
The slope of the tangentto the curve `x= t^2 + 3t - 8, y = 2t^2 - 2t - 5` at the point `(2, -1)` is
If the tangent to the curve y = x + siny at a point (a, b) is parallel to the line joining `(0, 3/2)` and `(1/2, 2)`, then ______.
If β is one of the angles between the normals to the ellipse, x2 + 3y2 = 9 at the points `(3cosθ, sqrt(3) sinθ)` and `(-3sinθ, sqrt(3) cos θ); θ ∈(0, π/2)`; then `(2 cot β)/(sin 2θ)` is equal to ______.
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 ______.
