Advertisements
Advertisements
प्रश्न
Find the angle between the rectangular hyperbola xy = 2 and the parabola x2 + 4y = 0
उत्तर
Given curves are xy = 2 ........(1)
x2 + 4y = 0 ........(2)
Now solving (1) and (2)
Substituting (1) in (2)
⇒ x2 + `4(2/x)` = 0
x3 + 8 = 0
x3 = – 8
x = – 2
Substituting in (1)
⇒ y = `2/(-2)` = – 1
∴ Point of intersection of (1) and (2) is (– 2, – 1)
xy = 2
⇒ y = `2/x` ........(1)
Differentiating w.r.t. ‘x’
`("d"y)/("d"x) = - 2/x^2`
Slope of the tangent 'm1' = `(("d"y)/("d"x))_(((-2, -1)))`
= `- 2/4 = - 1/2`
x2 + 4y = 0
⇒ y = `- x^2/4`
Differentiating w.r.t. 'x'
`("d"y)/("d"x) = - (2x)/4 = - x/2`
Slope of the tangent 'm2' = `(("d"y)/("d"x))_(((-2, -1)))`
= `2/2`
= 1
The angle between the curves
θ = `tan^-1 |("m"_1 - "m"_2)/(1 + "m"_1"m"_2)|`
θ = `tan^-1|((-1)/2 - 1)/(1 - 1/2)|`
`tan^-1|(- 3/2)/(1/2)|`
θ = `tan^1 (3)`
APPEARS IN
संबंधित प्रश्न
A particle moves along a straight line in such a way that after t seconds its distance from the origin is s = 2t2 + 3t metres. Find the average velocity between t = 3 and t = 6 seconds
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 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 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 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?
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 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 = x sin x at `(pi/2, 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
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:
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
Choose the correct alternative:
The maximum slope of the tangent to the curve y = ex sin x, x ∈ [0, 2π] is at
