Advertisements
Advertisements
प्रश्न
Find the condition for the following set of curve to intersect orthogonally \[\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1 \text { and } xy = c^2\] ?
Advertisements
उत्तर
\[ \frac{x^2}{a^2} - \frac{y^2}{b^2} = 1 . . . \left( 1 \right)\]
\[xy = c^2 . . . \left( 2 \right)\]
\[\text { Let the curves intersect orthogonally at }\left( x_1 , y_1 \right).\]
\[\text { On differentiating (1) on both sides w.r.t.x, we get }\]
\[\frac{2x}{a^2} - \frac{2y}{b^2}\frac{dy}{dx} = 0\]
\[ \Rightarrow \frac{dy}{dx} = \frac{x b^2}{a^2 y}\]
\[ \Rightarrow m_1 = \left( \frac{dy}{dx} \right)_\left( x_1 , y_1 \right) = \frac{x_1 b^2}{a^2 y_1}\]
\[\text { On differentiating (2) on both sides w.r.t.x, we get }\]
\[x\frac{dy}{dx} + y = 0\]
\[ \Rightarrow \frac{dy}{dx} = \frac{- y}{x}\]
\[ \Rightarrow m_2 = \left( \frac{dy}{dx} \right)_\left( x_1 , y_1 \right) = \frac{- y_1}{x_1}\]
\[\text { It is given that the curves intersect orhtogonally at }\left( x_1 , y_1 \right).\]
\[ \therefore m_1 \times m_2 = - 1\]
\[ \Rightarrow \frac{x_1 b^2}{a^2 y_1} \times \frac{- y_1}{x_1} = - 1\]
\[ \Rightarrow a^2 = b^2 \]
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.
The equation of tangent at (2, 3) on the curve y2 = ax3 + b is y = 4x – 5. Find the values of a and b.
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 (1, 3)
Find the equations of the tangent and normal to the given curves at the indicated points:
y = x3 at (1, 1)
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.]
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 equation of the normal to curve y2 = 4x at the point (1, 2).
Find the equations of the tangent and the normal, to the curve 16x2 + 9y2 = 145 at the point (x1, y1), where x1 = 2 and y1 > 0.
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 equation of the tangent and the normal to the following curve at the indicated point y = x4 − 6x3 + 13x2 − 10x + 5 at x = 1?
Find the equation of the tangent and the normal to the following curve at the indicated point y = x2 at (0, 0) ?
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 y2 = 4ax at \[\left( \frac{a}{m^2}, \frac{2a}{m} \right)\] ?
Find the equation of the tangent and the normal to the following curve at the indicated point 4x2 + 9y2 = 36 at (3cosθ, 2sinθ) ?
Find the equation of the tangent and the normal to the following curve at the indicated points x = at2, y = 2at at t = 1 ?
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 line to the curve y = x2 − 2x + 7 which is parallel to the line 2x − y + 9 = 0 ?
Prove that \[\left( \frac{x}{a} \right)^n + \left( \frac{y}{b} \right)^n = 2\] touches the straight line \[\frac{x}{a} + \frac{y}{b} = 2\] for all n ∈ N, at the point (a, b) ?
Show that the curves 2x = y2 and 2xy = k cut at right angles, if k2 = 8 ?
Prove that the curves xy = 4 and x2 + y2 = 8 touch each other ?
If the tangent line at a point (x, y) on the curve y = f(x) is parallel to x-axis, then write the value of \[\frac{dy}{dx}\] ?
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 ____________ .
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 tangent to the curve given by x = et . cost, y = et . sint at t = `pi/4` makes with x-axis an angle ______.
Find the angle of intersection of the curves y = 4 – x2 and y = x2.
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 tangent to the parabola x2 = 2y at the point (1, `1/2`) makes with the x-axis an angle of ____________.
The two curves x3 - 3xy2 + 5 = 0 and 3x2y - y3 - 7 = 0
The distance between the point (1, 1) and the tangent to the curve y = e2x + x2 drawn at the point x = 0
The tangent to the curve y = x2 + 3x will pass through the point (0, -9) if it is drawn at the point ____________.
The line y = x + 1 is a tangent to the curve y2 = 4x at the point
Find the equation of the tangent line to the curve y = x2 − 2x + 7 which is parallel to the line 2x − y + 9 = 0.
Let `y = f(x)` be the equation of the curve, then equation of normal is
Which of the following represent the slope of normal?
