Advertisements
Advertisements
Question
Show that the following curve intersect orthogonally at the indicated point x2 = 4y and 4y + x2 = 8 at (2, 1) ?
Solution
\[ x^2 = 4y . . . \left( 1 \right)\]
\[4y + x^2 = 8 . . . \left( 2 \right)\]
\[\text { Given point is }\left( 2, 1 \right)\]
\[\text { Differentiating (1) w.r.t.x,}\]
\[2x = 4\frac{dy}{dx}\]
\[ \Rightarrow \frac{dy}{dx} = \frac{x}{2}\]
\[ \Rightarrow m_1 = \left( \frac{dy}{dx} \right)_\left( 2, 1 \right) = \frac{2}{2} = 1\]
\[\text { Differentiating (2) w.r.t.x,}\]
\[4\frac{dy}{dx} + 2x = 0\]
\[ \Rightarrow \frac{dy}{dx} = \frac{- x}{2}\]
\[ \Rightarrow m_2 = \left( \frac{dy}{dx} \right)_\left( 2, 1 \right) = \frac{- 2}{2} = - 1\]
\[\text { Since, } m_1 \times m_2 = - 1\]
Hence, the given curves intersect orthogonally at the given point.
APPEARS IN
RELATED QUESTIONS
Find the point on the curve y = x3 − 11x + 5 at which the tangent is y = x − 11.
For the curve y = 4x3 − 2x5, find all the points at which the tangents passes through the origin.
Show that the normal at any point θ to the curve x = a cosθ + a θ sinθ, y = a sinθ – aθ cosθ is at a constant distance from the origin.
The slope of the tangent to the curve x = t2 + 3t – 8, y = 2t2 – 2t – 5 at the point (2,– 1) is
(A) `22/7`
(B) `6/7`
(C) `7/6`
(D) `(-6)/7`
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 = x3 − 2x2 − 2x at which the tangent lines are parallel to the line y = 2x− 3 ?
Find the points on the curve 2a2y = x3 − 3ax2 where the tangent is parallel to x-axis ?
Find the equation of the tangent to the curve \[\sqrt{x} + \sqrt{y} = a\] at the point \[\left( \frac{a^2}{4}, \frac{a^2}{4} \right)\] ?
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 \[x = \frac{2 a t^2}{1 + t^2}, y = \frac{2 a t^3}{1 + t^2}\text { at } t = \frac{1}{2}\] ?
Find the 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 a normal to the curve y = x loge x which is parallel to the line 2x − 2y + 3 = 0 ?
Find the equation of the tangent to the curve x2 + 3y − 3 = 0, which is parallel to the line y= 4x − 5 ?
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 ?
Find the angle of intersection of the following curve \[\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1\] and x2 + y2 = ab ?
Show that the following set of curve intersect orthogonally x3 − 3xy2 = −2 and 3x2y − y3 = 2 ?
Show that the following set of curve intersect orthogonally x2 + 4y2 = 8 and x2 − 2y2 = 4 ?
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 \[\frac{dy}{dx}\] ?
Find the slope of the normal at the point 't' on the curve \[x = \frac{1}{t}, y = t\] ?
Write the equation of the tangent drawn to the curve \[y = \sin x\] at the point (0,0) ?
The equation of the normal to the curve y = x + sin x cos x at x = `π/2` is ___________ .
If the tangent to the curve x = a t2, y = 2 at is perpendicular to x-axis, then its point of contact is _____________ .
At what point the slope of the tangent to the curve x2 + y2 − 2x − 3 = 0 is zero
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 a tangent and the normal to the curve `"y" = (("x" - 7))/(("x"-2)("x"-3)` at the point where it cuts the x-axis
Find the equation of all the tangents to the curve y = cos(x + y), –2π ≤ x ≤ 2π, that are parallel to the line x + 2y = 0.
Prove that the curves xy = 4 and x2 + y2 = 8 touch each other.
At what points on the curve x2 + y2 – 2x – 4y + 1 = 0, the tangents are parallel to the y-axis?
For which value of m is the line y = mx + 1 a tangent to the curve y2 = 4x?
`"sin"^"p" theta "cos"^"q" theta` attains a maximum, when `theta` = ____________.
Tangent is drawn to the ellipse `x^2/27 + y^2 = 1` at the point `(3sqrt(3) cos theta, sin theta), 0 < 0 < 1`. The sum of the intercepts on the axes made by the tangent is minimum if 0 is equal to
Tangent and normal are drawn at P(16, 16) on the parabola y2 = 16x, which intersect the axis of the parabola at A and B, respectively. If C is the centre of the circle through the points P, A and B and ∠CPB = θ, then a value of tan θ is:
The slope of the tangentto the curve `x= t^2 + 3t - 8, y = 2t^2 - 2t - 5` at the point `(2, -1)` is
