Advertisements
Advertisements
प्रश्न
The two curves x3 – 3xy2 + 2 = 0 and 3x2y – y3 – 2 = 0 intersect at an angle of ______.
विकल्प
`pi/4`
`pi/3`
`pi/2`
`pi/6`
Advertisements
उत्तर
The two curves x3 – 3xy2 + 2 = 0 and 3x2y – y3 – 2 = 0 intersect at an angle of `pi/2`.
Explanation:
The given curves are x3 – 3xy2 + 2 = 0 .....(i)
And 3x2y – y3 – 2 = 0 ......(ii)
Differentiating equation (i) w.r.t. x, we get
`3x^2 - 3 * (x * 2y "dy"/"dx" + y^2 * 1)` = 0
⇒ `x^2 - 2xy "dy"?'dx" - y^2` = 0
⇒ `2xy "dy"/"dx"` = x2 – y2
∴ `"dy"/"dx" = (x^2 - y^2)/(2xy)`
So slope of the curve m1 = `(x^2 - y^2)/(2xy)`
Differentiating equation (ii) w.r.t. x, we get
`3[x^2 "dy"/"dx" + y * 2x] - 3y^2 * "dy"/"dx"` = 0
`x^2 "dy"/"dx" + 2xy - y^2 "dy"/"dx"` = 0
⇒ `(x^3 - y^2) "dy"/"dx"` = – 2xy
∴ `"dy"/"dx" = (-2xy)/(x^2 - y^2)`
So the slope of the curve m2 = `(-2xy)/(x^2 - y^2)`
Now m1 × m2 = `(x^2 - y^2)/(2xy) xx (-2xy)/(x^2 - y^2)` = – 1
So the angle between the curves is `pi/2`.
APPEARS IN
संबंधित प्रश्न
Find the equations of the tangent and normal to the curve x = a sin3θ and y = a cos3θ at θ=π/4.
Prove that the least perimeter of an isosceles triangle in which a circle of radius r can be inscribed is `6sqrt3` r.
Find the slope of the tangent to the curve y = (x -1)/(x - 2), x != 2 at x = 10.
Find the slope of the normal to the curve x = acos3θ, y = asin3θ at `theta = pi/4`
Find points at which the tangent to the curve y = x3 − 3x2 − 9x + 7 is parallel to the x-axis.
The slope of the normal to the curve y = 2x2 + 3 sin x at x = 0 is
(A) 3
(B) 1/3
(C) −3
(D) `-1/3`
The line y = x + 1 is a tangent to the curve y2 = 4x at the point
(A) (1, 2)
(B) (2, 1)
(C) (1, −2)
(D) (−1, 2)
Find the points on the curve y = `4x^3 - 3x + 5` at which the equation of the tangent is parallel to the x-axis.
Find the slope of the tangent and the normal to the following curve at the indicted point x = a cos3 θ, y = a sin3 θ at θ = π/4 ?
Find the slope of the tangent and the normal to the following curve at the indicted point y = (sin 2x + cot x + 2)2 at x = π/2 ?
At what points on the curve y = x2 − 4x + 5 is the tangent perpendicular to the line 2y + x = 7?
Find the points on the curve\[\frac{x^2}{4} + \frac{y^2}{25} = 1\] at which the tangent is parallel to the y-axis ?
Find the points on the curve \[\frac{x^2}{9} + \frac{y^2}{16} = 1\] at which the tangent is parallel to x-axis ?
Find the equation of the tangent and the normal to the following curve at the indicated point xy = c2 at \[\left( ct, \frac{c}{t} \right)\] ?
Find the equation of the tangent and the normal to the following curve at the indicated point x2 = 4y at (2, 1) ?
Find the equation of the tangent and the normal to the following curve at the indicated points \[x = \frac{2 a t^2}{1 + t^2}, y = \frac{2 a t^3}{1 + t^2}\text { at } t = \frac{1}{2}\] ?
Find the equations of all lines of slope zero and that are tangent to the curve \[y = \frac{1}{x^2 - 2x + 3}\] ?
Show that the following set of curve intersect orthogonally y = x3 and 6y = 7 − x2 ?
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 ?
Write the value of \[\frac{dy}{dx}\] , if the normal to the curve y = f(x) at (x, y) is parallel to y-axis ?
If the tangent to a curve at a point (x, y) is equally inclined to the coordinates axes 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 on the tangent to the curve y = x2 − x + 2 at the point where it crosses the y-axis ?
The equation to the normal to the curve y = sin x at (0, 0) is ___________ .
The point on the curve y = x2 − 3x + 2 where tangent is perpendicular to y = x is ________________ .
The point on the curve y2 = x where tangent makes 45° angle with x-axis is ____________________ .
The equation of the normal to the curve 3x2 − y2 = 8 which is parallel to x + 3y = 8 is ____________ .
The slope of the tangent to the curve x = t2 + 3 t − 8, y = 2t2 − 2t − 5 at point (2, −1) is ________________ .
If the line y = x touches the curve y = x2 + bx + c at a point (1, 1) then _____________ .
The angle of intersection of the parabolas y2 = 4 ax and x2 = 4ay at the origin is ____________ .
The angle of intersection of the curves y = 2 sin2 x and y = cos 2 x at \[x = \frac{\pi}{6}\] is ____________ .
Any tangent to the curve y = 2x7 + 3x + 5 __________________ .
The two curves x3 – 3xy2 + 2 = 0 and 3x2y – y3 = 2 ______.
Find the angle of intersection of the curves y = 4 – x2 and y = x2.
For which value of m is the line y = mx + 1 a tangent to the curve y2 = 4x?
The line y = x + 1 is a tangent to the curve y2 = 4x at the point
The points at which the tangent passes through the origin for the curve y = 4x3 – 2x5 are
Two vertical poles of heights, 20 m and 80 m stand apart on a horizontal plane. The height (in meters) of the point of intersection of the lines joining the top of each pole to the foot of the other, From this horizontal plane is ______.
The normal of the curve given by the equation x = a(sinθ + cosθ), y = a(sinθ – cosθ) at the point θ is ______.
