Advertisements
Advertisements
Question
The equation of normal to the curve 3x2 – y2 = 8 which is parallel to the line x + 3y = 8 is ______.
Options
3x – y = 8
3x + y + 8 = 0
x + 3y ± 8 = 0
x + 3y = 0
Advertisements
Solution
The equation of normal to the curve 3x2 – y2 = 8 which is parallel to the line x + 3y = 8 is x + 3y ± 8 = 0.
Explanation:
Given equation of the curve is 3x2 – y2 = 8 ......(i)
Differentiating both sides w.r.t. x, we get
`6x - 2y * "dy"/"dx"` = 0
⇒ `"dy"/"dx" = (3x)/y`
`(3x)/y` is the slope of the tangent
∴ Slope of the normal = `(-1)/("dy"/"dx") = (-y)/(3x)`
Now x + 3y = 8 is parallel to the normal
Differentiating both sides w.r.t. x, we have
`1 + 3 "dy"/"dx"` = 0
⇒ `"dy"/"dx" = - 1/3`
∴ `(-y)/(3x) = - 1/3`
⇒ y = x
Putting y = x in equation (i) we get
3x2 – x2 = 8
⇒ 2x2 = 8
⇒ x2 = 4
∴ x = ± 2 and y = ± 2
So the points are (2, 2) and (– 2, – 2).
Equation of normal to the given curve at (2, 2) is
y – 2 = `- 1/3(x - 2)`
⇒ 3y – 6 = – x + 2
⇒ x + 3y – 8 = 0
Equation of normal at (– 2, – 2) is
y + 2 = `- 1/3 (x + 2)`
⇒ 3y + 6 = – x – 2
⇒ x + 3y + 8 = 0
∴ The equations of the normals to the curve are x + 3y ± 8 = 0.
APPEARS IN
RELATED QUESTIONS
Find the equation of the normal at a point on the curve x2 = 4y which passes through the point (1, 2). Also find the equation of the corresponding tangent.
Show that the equation of normal at any point t on the curve x = 3 cos t – cos3t and y = 3 sin t – sin3t is 4 (y cos3t – sin3t) = 3 sin 4t
Find the slope of the tangent to the curve y = 3x4 − 4x at x = 4.
Find the slope of the tangent to the curve y = x3 − 3x + 2 at the point whose x-coordinate is 3.
Find the equations of the tangent and normal to the given curves at the indicated points:
x = cos t, y = sin t at t = `pi/4`
Show that the tangents to the curve y = 7x3 + 11 at the points where x = 2 and x = −2 are parallel.
Find the points on the curve y = x3 at which the slope of the tangent is equal to the y-coordinate of the point.
Find the equation of the normal to curve y2 = 4x at the point (1, 2).
Show that the normal at any point θ to the curve x = a cosθ + a θ sinθ, y = a sinθ – aθ cosθ is at a constant distance from the origin.
Find the slope of the tangent and the normal to the following curve at the indicted point x2 + 3y + y2 = 5 at (1, 1) ?
Find the points on the curve x2 + y2 = 13, the tangent at each one of which is parallel to the line 2x + 3y = 7 ?
Find the equation of the tangent and the normal to the following curve at the indicated point\[\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \text{ at }\left( a\cos\theta, b\sin\theta \right)\] ?
Find the equation of the tangent and the normal to the following curve at the indicated point \[c^2 \left( x^2 + y^2 \right) = x^2 y^2 \text { at }\left( \frac{c}{\cos\theta}, \frac{c}{\sin\theta} \right)\] ?
Find the equation of the tangent line to the curve y = x2 + 4x − 16 which is parallel to the line 3x − y + 1 = 0 ?
Find the equation of the tangent to the curve \[y = \sqrt{3x - 2}\] which is parallel to the 4x − 2y + 5 = 0 ?
Find the equation of the tangent to the curve x = sin 3t, y = cos 2t at
\[t = \frac{\pi}{4}\] ?
Find the angle of intersection of the following curve \[\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1\] and x2 + y2 = ab ?
Find the angle of intersection of the following curve x2 + 4y2 = 8 and x2 − 2y2 = 2 ?
Show that the following set of curve intersect orthogonally x2 + 4y2 = 8 and x2 − 2y2 = 4 ?
If the straight line xcos \[\alpha\] +y sin \[\alpha\] = p touches the curve \[\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1\] then prove that a2cos2 \[\alpha\] \[-\] b2sin2 \[\alpha\] = p2 ?
If the tangent line at a point (x, y) on the curve y = f(x) is parallel to x-axis, then write the value of \[\frac{dy}{dx}\] ?
If the tangent line at a point (x, y) on the curve y = f(x) is parallel to y-axis, find the value of \[\frac{dx}{dy}\] ?
Write the equation of the normal to the curve y = x + sin x cos x at \[x = \frac{\pi}{2}\] ?
Write the angle between the curves y2 = 4x and x2 = 2y − 3 at the point (1, 2) ?
Write the coordinates of the point at which the tangent to the curve y = 2x2 − x + 1 is parallel to the line y = 3x + 9 ?
The equation to the normal to the curve y = sin x at (0, 0) is ___________ .
The equation of the normal to the curve y = x + sin x cos x at x = `π/2` is ___________ .
The point at the curve y = 12x − x2 where the slope of the tangent is zero will be _____________ .
The line y = mx + 1 is a tangent to the curve y2 = 4x, if the value of m is ________________ .
Find the equation of tangent to the curve y = x2 +4x + 1 at (-1 , -2).
The point on the curve y2 = x, where the tangent makes an angle of `pi/4` with x-axis is ______.
Find the co-ordinates of the point on the curve `sqrt(x) + sqrt(y)` = 4 at which tangent is equally inclined to the axes
The points at which the tangents to the curve y = x3 – 12x + 18 are parallel to x-axis are ______.
The tangent to the curve y = e2x at the point (0, 1) meets x-axis at ______.
The tangent to the parabola x2 = 2y at the point (1, `1/2`) makes with the x-axis an angle of ____________.
If `tan^-1x + tan^-1y + tan^-1z = pi/2`, then
Tangent and normal are drawn at P(16, 16) on the parabola y2 = 16x, which intersect the axis of the parabola at A and B, respectively. If C is the centre of the circle through the points P, A and B and ∠CPB = θ, then a value of tan θ is:
The points at which the tangent passes through the origin for the curve y = 4x3 – 2x5 are
An edge of variable cube is increasing at the rate of 3 cm/s. The volume of the cube increasing fast when the edge is 10 cm long is ______ cm3/s.
