Advertisements
Advertisements
प्रश्न
Show that the following curve intersect orthogonally at the indicated point x2 = y and x3 + 6y = 7 at (1, 1) ?
Advertisements
उत्तर
\[ x^2 = y . . . \left( 1 \right)\]
\[ x^3 + 6y = 7 . . . \left( 2 \right)\]
\[\text { Given point is }\left( 1, 1 \right)\]
\[\text { Differentiating (1) w.r.t.x, }\]
\[2x = \frac{dy}{dx}\]
\[ \Rightarrow m_1 = \left( \frac{dy}{dx} \right)_\left( 1, 1 \right) = 2\left( 1 \right) = 2\]
\[\text { Differentiating (2) w.r.t.x, }\]
\[3 x^2 + 6\frac{dy}{dx} = 0\]
\[ \Rightarrow \frac{dy}{dx} = \frac{- x^2}{2}\]
\[ \Rightarrow m_2 = \left( \frac{dy}{dx} \right)_\left( 1, 1 \right) = \frac{- 1}{2}\]
\[\text { Since,} m_1 \times m_2 = - 1\]
Hence, the given curves intersect orthogonally at the given point.
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.
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 point on the curve y = x3 − 11x + 5 at which the tangent is y = x − 11.
Find points on the curve `x^2/9 + "y"^2/16 = 1` at which the tangent is parallel to x-axis.
Find the equations of the tangent and normal to the given curves at the indicated points:
y = x4 − 6x3 + 13x2 − 10x + 5 at (0, 5)
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 values of a and b if the slope of the tangent to the curve xy + ax + by = 2 at (1, 1) is 2 ?
Find a point on the curve y = x3 − 3x where the tangent is parallel to the chord joining (1, −2) and (2, 2) ?
Find the points on the curve y2 = 2x3 at which the slope of the tangent is 3 ?
Find the points on the curve 2a2y = x3 − 3ax2 where the tangent is parallel to x-axis ?
Find the points on the curve \[\frac{x^2}{4} + \frac{y^2}{25} = 1\] at which the tangent is parallel to the x-axis ?
Find the equation of the tangent and the normal to the following curve at the indicated point \[\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1 \text { at } \left( a\sec\theta, b\tan\theta \right)\] ?
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 line to the curve y = x2 − 2x + 7 which perpendicular to the line 5y − 15x = 13. ?
Find the angle of intersection of the following curve x2 = 27y and y2 = 8x ?
Find the angle of intersection of the following curve x2 + y2 = 2x and y2 = x ?
Show that the following set of curve intersect orthogonally x3 − 3xy2 = −2 and 3x2y − y3 = 2 ?
Prove that the curves y2 = 4x and x2 + y2 - 6x + 1 = 0 touch each other at the point (1, 2) ?
Find the condition for the following set of curve to intersect orthogonally \[\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \text { and } \frac{x^2}{A^2} - \frac{y^2}{B^2} = 1\] ?
If the tangent to a curve at a point (x, y) is equally inclined to the coordinates axes then write the value of \[\frac{dy}{dx}\] ?
The angle of intersection of the curves xy = a2 and x2 − y2 = 2a2 is ______________ .
If the curves y = 2 ex and y = ae−x intersect orthogonally, then a = _____________ .
The point on the curve y = 6x − x2 at which the tangent to the curve is inclined at π/4 to the line x + y= 0 is __________ .
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 equation of tangents to the curve y = cos(x + y), –2π ≤ x ≤ 2π that are parallel to the line x + 2y = 0.
Find the equation of tangent to the curve `y = sqrt(3x -2)` which is parallel to the line 4x − 2y + 5 = 0. Also, write the equation of normal to the curve at the point of contact.
Find the angle of intersection of the curves y2 = x and x2 = y.
Prove that the curves y2 = 4x and x2 + y2 – 6x + 1 = 0 touch each other at the point (1, 2)
The equation of normal to the curve 3x2 – y2 = 8 which is parallel to the line x + 3y = 8 is ______.
The points at which the tangents to the curve y = x3 – 12x + 18 are parallel to x-axis are ______.
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
Find points on the curve `x^2/9 + "y"^2/16` = 1 at which the tangent is parallel to y-axis.
The Slope of the normal to the curve `y = 2x^2 + 3 sin x` at `x` = 0 is
If the tangent to the curve y = x + siny at a point (a, b) is parallel to the line joining `(0, 3/2)` and `(1/2, 2)`, then ______.
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 ______.
