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`
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 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.
Prove that the least perimeter of an isosceles triangle in which a circle of radius r can be inscribed is `6sqrt3` r.
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 equation of the tangent line to the curve y = x2 − 2x + 7 which is perpendicular to the line 5y − 15x = 13.
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 equations of the tangent and normal to the parabola y2 = 4ax at the point (at2, 2at).
Find the slope of the tangent and the normal to the following curve at the indicted point xy = 6 at (1, 6) ?
Find the points on the curve y2 = 2x3 at which the slope of the tangent is 3 ?
Find the points on the curve \[\frac{x^2}{9} + \frac{y^2}{16} = 1\] at which the tangent is parallel to y-axis ?
Who that the tangents to the curve y = 7x3 + 11 at the points x = 2 and x = −2 are parallel ?
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 and the normal to the following curve at the indicated point \[x^\frac{2}{3} + y^\frac{2}{3}\] = 2 at (1, 1) ?
Find an equation of normal line to the curve y = x3 + 2x + 6 which is 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 4x − 2y + 5 = 0 ?
At what points will be tangents to the curve y = 2x3 − 15x2 + 36x − 21 be parallel to x-axis ? Also, find the equations of the tangents to the curve at these points ?
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 ?
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 coordinates of the point on the curve y2 = x where the tangent line makes an angle \[\frac{\pi}{4}\] with x-axis ?
Write the equation on the tangent to the curve y = x2 − x + 2 at the point where it crosses the y-axis ?
The point on the curve y2 = x where tangent makes 45° angle with x-axis is ____________________ .
At what point the slope of the tangent to the curve x2 + y2 − 2x − 3 = 0 is zero
The angle of intersection of the curves xy = a2 and x2 − y2 = 2a2 is ______________ .
The points at which the tangents to the curve y = x3 – 12x + 18 are parallel to x-axis are ______.
Find a point on the curve y = (x – 2)2. at which the tangent is parallel to the chord joining the points (2, 0) and (4, 4).
If `tan^-1x + tan^-1y + tan^-1z = pi/2`, then
The number of common tangents to the circles x2 + y2 – 4x – 6x – 12 = 0 and x2 + y2 + 6x + 18y + 26 = 0 is
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 normals to the curve x = a(θ + sinθ), y = a(1 – cosθ) at the points θ = (2n + 1)π, n∈I are all ______.
