Advertisements
Advertisements
Question
Find the equation of the normals to the curve y = x3 + 2x + 6 which are parallel to the line x + 14y + 4 = 0.
Advertisements
Solution
The equation of the given curve is y = x3 + 2x + 6.
The slope of the tangent to the given curve at any point (x, y) is given by,
When x = 2, y = 8 + 4 + 6 = 18.
When x = −2, y = − 8 − 4 + 6 = −6.
Therefore, there are two normals to the given curve with slope -1/4 and passing through the points (2, 18) and (−2, −6).
Thus, the equation of the normal through (2, 18) is given by,
APPEARS IN
RELATED QUESTIONS
Show that the equation of normal at any point t on the curve x = 3 cos t – cos3t and y = 3 sin t – sin3t is 4 (y cos3t – sin3t) = 3 sin 4t
Find the slope of the tangent to the curve y = 3x4 − 4x at x = 4.
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 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.
Find the points on the curve x2 + y2 − 2x − 3 = 0 at which the tangents are parallel to the x-axis.
Find the equation of the tangent to the curve `y = sqrt(3x-2)` which is parallel to the line 4x − 2y + 5 = 0.
The slope of the tangent to the curve x = t2 + 3t – 8, y = 2t2 – 2t – 5 at the point (2,– 1) is
(A) `22/7`
(B) `6/7`
(C) `7/6`
(D) `(-6)/7`
The line y = mx + 1 is a tangent to the curve y2 = 4x if the value of m is
(A) 1
(B) 2
(C) 3
(D) 1/2
Find the points on the curve x2 + y2 = 13, the tangent at each one of which is parallel to the line 2x + 3y = 7 ?
At what points on the curve y = x2 − 4x + 5 is the tangent perpendicular to the line 2y + x = 7?
Find the points on the curve\[\frac{x^2}{4} + \frac{y^2}{25} = 1\] at which the tangent is parallel to the y-axis ?
Find the equation of the tangent and the normal to the following curve at the indicated point y = x2 + 4x + 1 at x = 3 ?
Find the equation of the tangent and the normal to the following curve at the indicated point \[c^2 \left( x^2 + y^2 \right) = x^2 y^2 \text { at }\left( \frac{c}{\cos\theta}, \frac{c}{\sin\theta} \right)\] ?
Find the equation of the tangent and the normal to the following curve at the indicated point y2 = 4x at (1, 2) ?
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 a normal to the curve y = x loge x which is parallel to the line 2x − 2y + 3 = 0 ?
Find the equation of the tangent line to the curve y = x2 − 2x + 7 which is parallel to the line 2x − y + 9 = 0 ?
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 equation of the tangents to the curve 3x2 – y2 = 8, which passes through the point (4/3, 0) ?
Find the angle of intersection of the following curve x2 = 27y and y2 = 8x ?
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 \[\frac{dy}{dx}\] ?
Find the slope of the normal at the point 't' on the curve \[x = \frac{1}{t}, y = t\] ?
Write the equation on the tangent to the curve y = x2 − x + 2 at the point where it crosses the y-axis ?
If the tangent to the curve x = a t2, y = 2 at is perpendicular to x-axis, then its point of contact is _____________ .
If the line y = x touches the curve y = x2 + bx + c at a point (1, 1) then _____________ .
Any tangent to the curve y = 2x7 + 3x + 5 __________________ .
The line y = mx + 1 is a tangent to the curve y2 = 4x, if the value of m is ________________ .
Find the angle of intersection of the curves \[y^2 = 4ax \text { and } x^2 = 4by\] .
Find the angle of intersection of the curves y2 = x and x2 = y.
The curve y = `x^(1/5)` has at (0, 0) ______.
For which value of m is the line y = mx + 1 a tangent to the curve y2 = 4x?
Tangents to the curve x2 + y2 = 2 at the points (1, 1) and (-1, 1) are ____________.
The line y = x + 1 is a tangent to the curve y2 = 4x at the point
Find the points on the curve `y = x^3` at which the slope of the tangent is equal to the y-coordinate of the point
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 curves y2 = 6x, 9x2 + by2 = 16, cut each other at right angles then the value of b 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 ______.
