Advertisements
Advertisements
Question
Find the equations of the tangents to the curve y = `- (x + 1)/(x - 1)` which are parallel to the line x + 2y = 6
Advertisements
Solution
Curse is y = `(x + 1)/(x - 1)`
DIfferentiating w.r.t. 'x'
`("d"y)/("d"x) = ((x - 1)(1) - (x + 1)(1))/(x - 1)^2`
Slope of the tangent 'm'
= `(x - 1 - x - 1)/(x - 1)^2`
= `- 2/(x - 1)^2`
Given line is x + 2y = 6
Slope of the line = ` 1/2`
Since the tangent is parallel to the line, then the slope of the tangent is `- 1/2`
∴ `("d"y)/("d"x) = 2/(x 1)^2 = - 1/2`
(x – 1)2 = 4
x – 1 = ± 2
x = – 1, 3
When x = – 1, y = 0
⇒ point is (– 1, 0)
When x = 3, y = 2
⇒ point is (3, 2)
Equation of tangent with slope `- 1/2` and at the point (– 1, 0) is
y – 0 = `- 1/2(x + 1)`
2y = – x – 1
⇒ x + 2y + 1 = 0
Equation of tangent with slope ` 1/2` and at the point (3, 2) is 2
y – 2 = `- 1/2 (x - 3)`
2y – 4 = – x + 3
x + 2y – 7 = 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 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 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 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 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?
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 curve y = x3 – 6x2 + x + 3 where the normal is parallel to the line x + y = 1729
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
y = x sin x at `(pi/2, 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
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:
The position of a particle moving along a horizontal line of any time t is given by s(t) = 3t2 – 2t – 8. The time at which the particle is at rest is
Choose the correct alternative:
Find the point on the curve 6y = x3 + 2 at which y-coordinate changes 8 times as fast as x-coordinate 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
