Advertisements
Advertisements
प्रश्न
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
Advertisements
उत्तर
x = 7 cos t and y = 2 sin t, t ∈ R
Differentiating w.r.t. ‘t’,
`("d"x)/"dt"` = – 7 sin t and `("d"y)/"dt"` = 2 cos t
Slope of the tangent ‘m’
`("d"y)/("d"x) = (("d"y)/("dt"))/(("d"x)/("d"t"))`
= `(2 cot"t")/(- 7 sin "t")`
Any point on the curve is (7 Cos t, 2 sin t)
Equation of tangent is y – y1 = m (x – x1)
y – 2 sint = `- (2cot"t")/(7sin"t")` (x – 7 cos t)
7y sin t – 14 sin2t = – 2x cos t + 14 cos2t
2x cos t + 7 y sin t – 14(sin2t + cos2t) = 0
2x cos t + 7y sin t – 14 = 0
Now slope of normal is `- 1/3 = (7sin"t")/(2cos"t")`
Equation of normal is y – y1 = `- 1/"m"` (x – x1)
y – 2 sin t = `(7sin"t")/(2cos"t")` (x – 7 cos t)
2y cos t – 4 sin t cos t = 7x sin t – 49 sin t cos t 7x sin t – 2y cos t – 45 sin t cos t = 0
APPEARS IN
संबंधित प्रश्न
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 instantaneous velocity of the camera when it hits the ground?
A particle moves along a line according to the law s(t) = 2t3 – 9t2 + 12t – 4, where t ≥ 0. At what times the particle changes direction?
A beacon makes one revolution every 10 seconds. It is located on a ship which is anchored 5 km from a straight shoreline. How fast is the beam moving along the shoreline when it makes an angle of 45° with the shore?
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 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, at what rate, the area of the triangle formed by the ladder, wall, and the floor, is changing?
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 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
y = x4 + 2ex at (0, 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
Show that the two curves x2 – y2 = r2 and xy = c2 where c, r are constants, cut orthogonally
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:
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
