Advertisements
Advertisements
Question
Find the points on curve y = x3 – 6x2 + x + 3 where the normal is parallel to the line x + y = 1729
Advertisements
Solution
y = x3 – 6x2 + x + 3
Differentiating w.r.t. ‘x’
Slope of the tangent `("d"y)/("d"x)` = 3x2 – 12x + 1
Slope of the normal = `1/(3x^2 - 12x + 1)`
Given line is x + y = 1729
Slope of the line is – 1
Since the normal is parallel to the line, their slopes are equal.
`1/(3x^2 - 12x + 1)` = – 1
3x2 – 12x + 1 = 1
3x2 – 12x = 0
3x(x – 4) = 0
x = 0, 4
When x = 0, y = (0)3 – 6(0)2 + 0 + 3 = 3
When x = 4, y = (4)3 – 6(4)2 + 4 + 3
= 64 – 96 + 4 + 3
= – 25
∴ The points on the curve are (0, 3) and (4, – 25).
APPEARS IN
RELATED QUESTIONS
A camera is accidentally knocked off an edge of a cliff 400 ft high. The camera falls a distance of s = 16t2 in t seconds. How long does the camera fall before it hits the ground?
A particle moves along a line according to the law s(t) = 2t3 – 9t2 + 12t – 4, where t ≥ 0. Find the total distance travelled by the particle in the first 4 seconds
If the volume of a cube of side length x is v = x3. Find the rate of change of the volume with respect to x when x = 5 units
A conical water tank with vertex down of 12 metres height has a radius of 5 metres at the top. If water flows into the tank at a rate 10 cubic m/min, how fast is the depth of the water increases when the water is 8 metres deep?
A police jeep, approaching an orthogonal intersection from the northern direction, is chasing a speeding car that has turned and moving straight east. When the jeep is 0.6 km north of the intersection and the car is 0.8 km to the east. The police determine with a radar that the distance between them and the car is increasing at 20 km/hr. If the jeep is moving at 60 km/hr at the instant of measurement, what is the speed of the car?
Find the slope of the tangent to the following curves at the respective given points.
y = x4 + 2x2 – x at x = 1
Find the slope of the tangent to the following curves at the respective given points.
x = a cos3t, y = b sin3t at t = `pi/2`
Find the points on the curve y2 – 4xy = x2 + 5 for which the tangent is horizontal
Find the tangent and normal to the following curves at the given points on the curve
x = cos t, y = 2 sin2t at t = `pi/2`
Find the equations of the tangents to the curve y = 1 + x3 for which the tangent is orthogonal with the line x + 12y = 12
Find the angle between the rectangular hyperbola xy = 2 and the parabola x2 + 4y = 0
Show that the two curves x2 – y2 = r2 and xy = c2 where c, r are constants, cut orthogonally
Choose the correct alternative:
A stone is thrown, up vertically. The height reaches at time t seconds is given by x = 80t – 16t2. The stone reaches the maximum! height in time t seconds is given by
Choose the correct alternative:
The abscissa of the point on the curve f(x) = `sqrt(8 - 2x)` at which the slope of the tangent is – 0.25?
Choose the correct alternative:
The tangent to the curve y2 – xy + 9 = 0 is vertical when
Choose the correct alternative:
Angle between y2 = x and x2 = y at the origin is
