Advertisements
Advertisements
Question
Find the equations of the tangents to the curve y = 1 + x3 for which the tangent is orthogonal with the line x + 12y = 12
Advertisements
Solution
Curve y = 1 + x3
Differentiating w.r.t ‘x’
Slope of the tangent ‘m’ = `("d"y)/("d"x)` = 3x2
Given line is x + 12y = 12
Slope of the line is `- 1/12`
Since the tangent is orthogonal with the line, the slope of the tangent is 12.
∴ `("d"y)/("d"x)` = 12
i.e 3x2 = 12
x2 = 4
x = ± 2
When x = 2
y = 1 + 8
= 9
⇒ point is (2, 9)
When x = – 2
y = 1 – 8
= – 7
⇒ point is (– 2, – 7)
Equation of tangent with slope 12 and at the j point (2, 9) is
y – 9 = 12(x – 2)
y – 9 = 12x – 24
12x – y – 15 = 0
Equation of tangent with slope 12 and at the point (– 2, – 7) is
y + 7 = 12(x + 2)
y + 7 = 12x + 24
12x – y + 17 = 0
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 camera is accidentally knocked off an edge of a cliff 400 ft high. The camera falls a distance of s = 16t2 in t seconds. What is the average velocity with which the camera falls during the last 2 seconds?
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
A stone is dropped into a pond causing ripples in the form of concentric circles. The radius r of the outer ripple is increasing at a constant rate at 2 cm per second. When the radius is 5 cm find the rate of changing of the total area of the disturbed water?
A ladder 17 metre long is leaning against the wall. The base of the ladder is pulled away from the wall at a rate of 5 m/s. When the base of the ladder is 8 metres from the wall. How fast is the top of the ladder moving down the wall?
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.
x = a cos3t, y = b sin3t at t = `pi/2`
Find the point on the curve y = x2 – 5x + 4 at which the tangent is parallel to the line 3x + y = 7
Find the tangent and normal to the following curves at the given points on the curve
y = x2 – x4 at (1, 0)
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 equation of tangent and normal to the curve given by x – 7 cos t andy = 2 sin t, t ∈ R at any point on the curve
Find the angle between the rectangular hyperbola xy = 2 and the parabola x2 + 4y = 0
Choose the correct alternative:
The volume of a sphere is increasing in volume at the rate of 3π cm3/ sec. The rate of change of its radius when radius is `1/2` cm
Choose the correct alternative:
A balloon rises straight up at 10 m/s. An observer is 40 m away from the spot where the balloon left the ground. The rate of change of the balloon’s angle of elevation in radian per second when the balloon is 30 metres above the ground
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 slope of the line normal to the curve f(x) = 2 cos 4x at x = `pi/12` is
Choose the correct alternative:
Angle between y2 = x and x2 = y at the origin is
Choose the correct alternative:
The maximum slope of the tangent to the curve y = ex sin x, x ∈ [0, 2π] is at
