Advertisements
Advertisements
Question
Find the equation of the tangent and the normal to the following curve at the indicated point y = x2 + 4x + 1 at x = 3 ?
Advertisements
Solution
\[y = x^2 + 4x + 1\]
\[\text { Differentiating both sides w.r.t.x, } \]
\[\frac{dy}{dx} = 2x + 4\]
\[\text { When x}=3,y = 9 + 12 + 1 = 22\]
\[\text { So }, \left( x_1 , y_1 \right) = \left( 3, 22 \right)\]
\[\text { Slope of tangent },m= \left( \frac{dy}{dx} \right)_{x = 3} =10\]
\[\text { Equation of tangent is },\]
\[y - y_1 = m \left( x - x_1 \right)\]
\[ \Rightarrow y - 22 = 10\left( x - 3 \right)\]
\[ \Rightarrow y - 22 = 10x - 30\]
\[ \Rightarrow 10x - y - 8 = 0\]
\[\text { Equation of normal is },\]
\[y - y_1 = \frac{- 1}{m} \left( x - x_1 \right)\]
\[ \Rightarrow y - 22 = \frac{- 1}{10} \left( x - 3 \right)\]
\[ \Rightarrow 10y - 220 = - x + 3\]
\[ \Rightarrow x + 10y - 223 = 0\]
APPEARS IN
RELATED QUESTIONS
Find the slope of the tangent to the curve y = 3x4 − 4x at x = 4.
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 equation of all lines having slope 2 which are tangents to the curve `y = 1/(x- 3), x != 3`
For the curve y = 4x3 − 2x5, find all the points at which the tangents passes through the origin.
Find the equation of the normals to the curve y = x3 + 2x + 6 which are parallel to the line x + 14y + 4 = 0.
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 equation of the normal to curve y2 = 4x at the point (1, 2).
Find the equations of the tangent and the normal, to the curve 16x2 + 9y2 = 145 at the point (x1, y1), where x1 = 2 and y1 > 0.
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 x = a (θ − sin θ), y = a(1 − cos θ) at θ = π/2 ?
Find the points on the curve xy + 4 = 0 at which the tangents are inclined at an angle of 45° with the x-axis ?
Find the point on the curve y = x2 where the slope of the tangent is equal to the x-coordinate of the point ?
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 normal to y = 2x3 − x2 + 3 at (1, 4) ?
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 \[x^\frac{2}{3} + y^\frac{2}{3}\] = 2 at (1, 1) ?
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 tangent and the normal to the following curve at the indicated points x = asect, y = btant at t ?
Find the equations of all lines of slope zero and that are tangent to the curve \[y = \frac{1}{x^2 - 2x + 3}\] ?
Find the equation of the tangent to the curve x2 + 3y − 3 = 0, which is parallel to the line y= 4x − 5 ?
Find the angle of intersection of the following curve y = x2 and x2 + y2 = 20 ?
Find the angle of intersection of the following curve 2y2 = x3 and y2 = 32x ?
Find the angle of intersection of the following curve x2 + 4y2 = 8 and x2 − 2y2 = 2 ?
The point at the curve y = 12x − x2 where the slope of the tangent is zero will be _____________ .
The equation of the normal to the curve 3x2 − y2 = 8 which is parallel to x + 3y = 8 is ____________ .
The angle of intersection of the parabolas y2 = 4 ax and x2 = 4ay at the origin is ____________ .
The slope of the tangent to the curve x = t2 + 3t − 8, y = 2t2 − 2t − 5 at the point (2, −1) is _____________ .
The normal at the point (1, 1) on the curve 2y + x2 = 3 is _____________ .
The normal to the curve x2 = 4y passing through (1, 2) is _____________ .
Find the angle of intersection of the curves y2 = 4ax and x2 = 4by.
Prove that the curves xy = 4 and x2 + y2 = 8 touch each other.
Find the angle of intersection of the curves y = 4 – x2 and y = x2.
The equation of normal to the curve 3x2 – y2 = 8 which is parallel to the line x + 3y = 8 is ______.
At (0, 0) the curve y = x3 + x
The two curves x3 - 3xy2 + 5 = 0 and 3x2y - y3 - 7 = 0
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 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 ______.
If m be the slope of a tangent to the curve e2y = 1 + 4x2, then ______.
