Advertisements
Advertisements
Question
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.
Advertisements
Solution
Slope of the given line is 2
Let (x1, y1) be the point where the tangent is drawn to the curve `y = sqrt(3x -2)`
Since, the point lies on the curve.
Hence, `y_1 = sqrt(3x_1-2)` ...(1)
Now, `y = sqrt(3x -2)`
⇒ `dy/dx = 3/(2sqrt(3x-2)`
Slope of tangent at `(x_1,y_1) = 3/(2sqrt(3x_1-2)`
Given that
Slope of tangent = slope of the given line
⇒ `3/(2sqrt(3x_1-2)` = 2
⇒ `3 = 4 sqrt(3x_1-2)`
⇒ `9 = 16 (3x_1 - 2)`
⇒ `9/16 = 3x_1 -2`
⇒ `3x_1 = 9/16 + 2 = (9+32)/16 = 41/16`
⇒ `x_1 = 41/48`
Now,
`y_1 = sqrt(123/48 -2) = sqrt(27/48) = sqrt(9/16) = 3/4` ...[From (1)]
∴ `(x_1,y_1) = (41/48,3/4)`
Equation of tangent is,
`y - y_1 = m (x -x_1)`
⇒ `y - 3/4 = 2 (x-41/48)`
⇒ `(4y-3)/4 = 2 ((48x -41)/48)`
⇒ 24y - 18 = 48x - 41
⇒ 48x - 24y - 23 = 0
Equation of normal at the point of contact will be
`y-y_1 = -1/m (x -x_1)`
⇒ `y - 3/4 = -1/2 (x -41/48)`
⇒ `(4y-3)/4 = -1/2 (x -41/48)`
⇒ `(4y-3)/2 = (41/48 -x)`
⇒ `(4y-3)/2 = (41-48x)/48`
⇒ `4y - 3 = (41 -48x)/24`
⇒ 96y - 72 = 41 - 48x
⇒ 48x + 96y = 113
RELATED QUESTIONS
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.
Find the equations of the tangent and normal to the given curves at the indicated points:
y = x4 − 6x3 + 13x2 − 10x + 5 at (0, 5)
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 x2 + y2 − 2x − 3 = 0 at which the tangents are parallel to the x-axis.
Find the equations of the tangent and normal to the parabola y2 = 4ax at the point (at2, 2at).
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 equation of the normal to curve y2 = 4x at the point (1, 2).
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 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 points x = at2, y = 2at at t = 1 ?
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 \[\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1\] and x2 + y2 = ab ?
Write the angle between the curves y = e−x and y = ex at their point of intersections ?
Write the equation of the normal to the curve y = cos x at (0, 1) ?
The equation of the normal to the curve y = x + sin x cos x at x = `π/2` is ___________ .
The equation of the normal to the curve y = x(2 − x) at the point (2, 0) is ________________ .
The angle between the curves y2 = x and x2 = y at (1, 1) is ______________ .
The normal at the point (1, 1) on the curve 2y + x2 = 3 is _____________ .
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.
Prove that the curves y2 = 4x and x2 + y2 – 6x + 1 = 0 touch each other at the point (1, 2)
The points at which the tangents to the curve y = x3 – 12x + 18 are parallel to x-axis are ______.
The slope of tangent to the curve x = t2 + 3t – 8, y = 2t2 – 2t – 5 at the point (2, –1) is ______.
The equation of normal to the curve y = tanx at (0, 0) is ______.
The point on the curves y = (x – 3)2 where the tangent is parallel to the chord joining (3, 0) and (4, 1) is ____________.
The points at which the tangent passes through the origin for the curve y = 4x3 – 2x5 are
The Slope of the normal to the curve `y = 2x^2 + 3 sin x` at `x` = 0 is
If (a, b), (c, d) are points on the curve 9y2 = x3 where the normal makes equal intercepts on the axes, then the value of a + b + c + d is ______.
