Advertisements
Advertisements
प्रश्न
Prove that the curves x = y2 and xy = k cut at right angles if 8k2 = 1. [Hint: Two curves intersect at right angle if the tangents to the curves at the point of intersection are perpendicular to each other.]
Advertisements
उत्तर
The equations of the given curves are given as `x = y^2 and xy = k`
Putting x = y2 in xy = k, we get:
This implies that we should have the product of the tangents as − 1.
Thus, the given two curves cut at right angles if the product of the slopes of their respective tangents at
APPEARS IN
संबंधित प्रश्न
Find the slope of the tangent to the curve y = x3 − 3x + 2 at the point whose x-coordinate is 3.
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 normals to the curve y = x3 + 2x + 6 which are parallel to the line x + 14y + 4 = 0.
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)
If the tangent to the curve y = x3 + ax + b at (1, − 6) is parallel to the line x − y + 5 = 0, find a and b ?
Find the points on the curve xy + 4 = 0 at which the tangents are inclined at an angle of 45° with the x-axis ?
At what points on the circle x2 + y2 − 2x − 4y + 1 = 0, the tangent is parallel to x-axis?
Find the points on the curve y = 3x2 − 9x + 8 at which the tangents are equally inclined with the axes ?
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 y = x4 − bx3 + 13x2 − 10x + 5 at (0, 5) ?
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 x2 = 4y at (2, 1) ?
Find the equation of the tangent to the curve x = θ + sin θ, y = 1 + cos θ at θ = π/4 ?
Find the equation of the tangent and the normal to the following curve at the indicated points x = θ + sinθ, y = 1 + cosθ at θ = \[\frac{\pi}{2}\] ?
Find the equation of the normal to the curve x2 + 2y2 − 4x − 6y + 8 = 0 at the point whose abscissa is 2 ?
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 x = sin 3t, y = cos 2t at
\[t = \frac{\pi}{4}\] ?
Find the angle of intersection of the following curve y2 = x and x2 = y ?
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 \[\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1 \text { and } xy = c^2\] ?
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 line at a point (x, y) on the curve y = f(x) is parallel to y-axis, find the value of \[\frac{dx}{dy}\] ?
Find the slope of the normal at the point 't' on the curve \[x = \frac{1}{t}, y = t\] ?
Write the angle made by the tangent to the curve x = et cos t, y = et sin t at \[t = \frac{\pi}{4}\] with the x-axis ?
Write the equation of the normal to the curve y = cos x at (0, 1) ?
The equation to the normal to the curve y = sin x at (0, 0) is ___________ .
If the tangent to the curve x = a t2, y = 2 at is perpendicular to x-axis, then its point of contact is _____________ .
If the line y = x touches the curve y = x2 + bx + c at a point (1, 1) then _____________ .
The slope of the tangent to the curve x = 3t2 + 1, y = t3 −1 at x = 1 is ___________ .
If the curves y = 2 ex and y = ae−x intersect orthogonally, then a = _____________ .
The slope of the tangent to the curve x = t2 + 3t − 8, y = 2t2 − 2t − 5 at the point (2, −1) is _____________ .
The line y = mx + 1 is a tangent to the curve y2 = 4x, if the value of m is ________________ .
Find the angle of intersection of the curves \[y^2 = 4ax \text { and } x^2 = 4by\] .
The two curves x3 – 3xy2 + 2 = 0 and 3x2y – y3 = 2 ______.
The equation of tangent to the curve y(1 + x2) = 2 – x, where it crosses x-axis is ______.
At (0, 0) the curve y = x3 + x
The number of common tangents to the circles x2 + y2 – 4x – 6x – 12 = 0 and x2 + y2 + 6x + 18y + 26 = 0 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.
