Advertisements
Advertisements
Question
Find the angle of intersection of the following curve x2 = 27y and y2 = 8x ?
Advertisements
Solution
\[\text { Given curves are },\]
\[ x^2 = 27y . . . \left( 1 \right)\]
\[ y^2 = 8x . . . \left( 2 \right)\]
\[\text { From } (2) \text { we get }\]
\[x = \frac{y^2}{8} \]
\[\text { Substituting this in }(1),\]
\[ \left( \frac{y^2}{8} \right)^2 = 27y\]
\[ \Rightarrow y^4 = 1728y\]
\[ \Rightarrow y \left( y^3 - {12}^3 \right) = 0\]
\[ \Rightarrow y = 0 ory = 12\]
\[\text { Substituting the values of y in (2), we get }, \]
\[ \Rightarrow x = 0 orx = 18\]
\[ \Rightarrow \left( x, y \right)=\left( 0, 0 \right),\left( 18, 12 \right)\]
\[\text { Differentiating (1) w.r.t.x },\]
\[2x = 27\frac{dy}{dx}\]
\[ \Rightarrow \frac{dy}{dx} = \frac{2x}{27} . . . \left( 3 \right)\]
\[\text { Differentiating (2) w.r.t.x },\]
\[2y \frac{dy}{dx} = 8\]
\[ \Rightarrow \frac{dy}{dx} = \frac{4}{y} . . . \left( 4 \right)\]
\[\text { Case } - 1:\left( x, y \right)=\left( 0, 0 \right)\]
\[\text { From }\left( 4 \right) \text { we have,} m_2 \text { is undefined }\]
\[ \therefore\text { We cannot find } \theta\]
\[\text { Case -} 2: \left( x, y \right)=\left( 18, 12 \right)\]
\[\text { From } \left( 3 \right) \text { we have }, m_1 = \frac{36}{27} = \frac{4}{3}\]
\[\text { From } \left( 4 \right) \text { we have }, m_2 = \frac{4}{12} = \frac{1}{3}\]
\[\text { Now }, \]
\[\tan \theta = \left| \frac{m_1 - m_2}{1 + m_1 m_2} \right| = \left| \frac{\frac{4}{3} - \frac{1}{3}}{1 + \frac{4}{9}} \right| = \frac{9}{13}\]
\[ \Rightarrow \theta = \tan^{- 1} \left( \frac{9}{13} \right)\]
APPEARS IN
RELATED QUESTIONS
Find the equations of the tangent and normal to the curve `x^2/a^2−y^2/b^2=1` at the point `(sqrt2a,b)` .
Find the slope of the tangent to curve y = x3 − x + 1 at the point whose x-coordinate is 2.
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).
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.
Find the slope of the tangent and the normal to the following curve at the indicted point \[y = \sqrt{x^3} \text { at } x = 4\] ?
Find the slope of the tangent and the normal to the following curve at the indicted point y = 2x2 + 3 sin x at x = 0 ?
Find the slope of the tangent and the normal to the following curve at the indicted point x = a cos3 θ, y = a sin3 θ at θ = π/4 ?
Find the slope of the tangent and the normal to the following curve at the indicted point xy = 6 at (1, 6) ?
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 the points on the curve y = 3x2 − 9x + 8 at which the tangents are equally inclined with the axes ?
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 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 \[y^2 = \frac{x^3}{4 - x}at \left( 2, - 2 \right)\] ?
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 xy = c2 at \[\left( ct, \frac{c}{t} \right)\] ?
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 = θ + sinθ, y = 1 + cosθ at θ = \[\frac{\pi}{2}\] ?
Find the equation of the tangent and the normal to the following curve at the indicated points x = at2, y = 2at at t = 1 ?
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 equation of the tangents to the curve 3x2 – y2 = 8, which passes through the point (4/3, 0) ?
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 curve intersect orthogonally at the indicated point x2 = 4y and 4y + x2 = 8 at (2, 1) ?
Find the slope of the normal at the point 't' on the curve \[x = \frac{1}{t}, y = t\] ?
The point on the curve y2 = x where tangent makes 45° angle with x-axis is ____________________ .
Any tangent to the curve y = 2x7 + 3x + 5 __________________ .
The normal at the point (1, 1) on the curve 2y + x2 = 3 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.
Show that the equation of normal at any point on the curve x = 3cos θ – cos3θ, y = 3sinθ – sin3θ is 4 (y cos3θ – x sin3θ) = 3 sin 4θ
The tangent to the curve given by x = et . cost, y = et . sint at t = `pi/4` makes with x-axis an angle ______.
At what points on the curve x2 + y2 – 2x – 4y + 1 = 0, the tangents are parallel to the y-axis?
The curve y = `x^(1/5)` has at (0, 0) ______.
The equation of tangent to the curve y(1 + x2) = 2 – x, where it crosses x-axis is ______.
The tangent to the parabola x2 = 2y at the point (1, `1/2`) makes with the x-axis an angle of ____________.
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
The number of common tangents to the circles x2 + y2 – 4x – 6x – 12 = 0 and x2 + y2 + 6x + 18y + 26 = 0 is
The line is y = x + 1 is a tangent to the curve y2 = 4x at the point.
The number of values of c such that the straight line 3x + 4y = c touches the curve `x^4/2` = x + y is ______.
If β is one of the angles between the normals to the ellipse, x2 + 3y2 = 9 at the points `(3cosθ, sqrt(3) sinθ)` and `(-3sinθ, sqrt(3) cos θ); θ ∈(0, π/2)`; then `(2 cot β)/(sin 2θ)` is equal to ______.
