Advertisements
Advertisements
प्रश्न
The equation of normal to the curve 3x2 – y2 = 8 which is parallel to the line x + 3y = 8 is ______.
विकल्प
3x – y = 8
3x + y + 8 = 0
x + 3y ± 8 = 0
x + 3y = 0
Advertisements
उत्तर
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
संबंधित प्रश्न
The equation of tangent at (2, 3) on the curve y2 = ax3 + b is y = 4x – 5. Find the values of a and b.
Find the equation of the tangent line to the curve y = x2 − 2x + 7 which is perpendicular to the line 5y − 15x = 13.
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 points on the curve x2 + y2 − 2x − 3 = 0 at which the tangents are parallel to the x-axis.
Find the equation of the normal at the point (am2, am3) for the curve ay2 = x3.
Find the equation of the normals to the curve y = x3 + 2x + 6 which are parallel to the line x + 14y + 4 = 0.
Find the equation of the tangent to the curve `y = sqrt(3x-2)` which is parallel to the line 4x − 2y + 5 = 0.
Find the slope of the tangent and the normal to the following curve at the indicted point x = a (θ − sin θ), y = a(1 − cos θ) at θ = π/2 ?
Find the points on the curve y = 3x2 − 9x + 8 at which the tangents are equally inclined with the axes ?
Find the point on the curve y = 3x2 + 4 at which the tangent is perpendicular to the line whose slop is \[- \frac{1}{6}\] ?
Find the equation of the tangent and the normal to the following curve at the indicated point \[y^2 = \frac{x^3}{4 - x}at \left( 2, - 2 \right)\] ?
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 tangents to the curve 3x2 – y2 = 8, which passes through the point (4/3, 0) ?
Find the angle of intersection of the following curve y = x2 and x2 + y2 = 20 ?
Find the angle of intersection of the following curve x2 + y2 = 2x and y2 = x ?
Show that the following curve intersect orthogonally at the indicated point y2 = 8x and 2x2 + y2 = 10 at \[\left( 1, 2\sqrt{2} \right)\] ?
Show that the curves 2x = y2 and 2xy = k cut at right angles, if k2 = 8 ?
Find the condition for the following set of curve to intersect orthogonally \[\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1 \text { and } xy = c^2\] ?
Find the slope of the tangent to the curve x = t2 + 3t − 8, y = 2t2 − 2t − 5 at t = 2 ?
The equations of tangent at those points where the curve y = x2 − 3x + 2 meets x-axis are _______________ .
If the curve ay + x2 = 7 and x3 = y cut orthogonally at (1, 1), then a is equal to _____________ .
If the curves y = 2 ex and y = ae−x intersect orthogonally, then a = _____________ .
The angle of intersection of the curves y = 2 sin2 x and y = cos 2 x at \[x = \frac{\pi}{6}\] is ____________ .
The normal to the curve x2 = 4y passing through (1, 2) is _____________ .
Show that the equation of normal at any point on the curve x = 3cos θ – cos3θ, y = 3sinθ – sin3θ is 4 (y cos3θ – x sin3θ) = 3 sin 4θ
The point on the curve y2 = x, where the tangent makes an angle of `pi/4` with x-axis is ______.
The two curves x3 – 3xy2 + 2 = 0 and 3x2y – y3 – 2 = 0 intersect at an angle of ______.
The point on the curves y = (x – 3)2 where the tangent is parallel to the chord joining (3, 0) and (4, 1) is ____________.
The two curves x3 - 3xy2 + 5 = 0 and 3x2y - y3 - 7 = 0
The distance between the point (1, 1) and the tangent to the curve y = e2x + x2 drawn at the point x = 0
Find the points on the curve `y = x^3` at which the slope of the tangent is equal to the y-coordinate of the point
The slope of the tangentto the curve `x= t^2 + 3t - 8, y = 2t^2 - 2t - 5` at the point `(2, -1)` is
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.
If the tangent to the conic, y – 6 = x2 at (2, 10) touches the circle, x2 + y2 + 8x – 2y = k (for some fixed k) at a point (α, β); then (α, β) is ______.
