Advertisements
Advertisements
Question
Write the angle between the curves y2 = 4x and x2 = 2y − 3 at the point (1, 2) ?
Advertisements
Solution
\[\text { Given }: \]
\[ y^2 = 4x . . . \left( 1 \right)\]
\[ x^2 = 2y - 3 . . . \left( 2 \right)\]
\[\text { On differentiating (1) w.r.t.x, we get }\]
\[2y\frac{dy}{dx} = 4\]
\[ \Rightarrow \frac{dy}{dx} = \frac{2}{y}\]
\[ \Rightarrow m_1 = \left( \frac{dy}{dx} \right)_\left( 1, 2 \right) = \frac{2}{2} = 1\]
\[\text { On differentiating (2) w.r.t.x, we get }\]
\[2x = 2\frac{dy}{dx}\]
\[ \Rightarrow \frac{dy}{dx} = x\]
\[ \Rightarrow m_2 = \left( \frac{dy}{dx} \right)_\left( 1, 2 \right) = 1\]
\[\text { Thus, we get }\]
\[\tan \theta = \left| \frac{m_1 - m_2}{1 + m_1 m_2} \right|\]
\[ \Rightarrow \tan \theta = \left| \frac{1 - 1}{1 + 1} \right|\]
\[ \Rightarrow \tan \theta = 0\]
\[ \Rightarrow \theta = 0^o\]
APPEARS IN
RELATED QUESTIONS
Find the slope of the tangent to the curve y = 3x4 − 4x at x = 4.
Find the point on the curve y = x3 − 11x + 5 at which the tangent is y = x − 11.
Find the equations of all lines having slope 0 which are tangent to the curve y = `1/(x^2-2x + 3)`
Find the slope of the tangent and the normal to the following curve at the indicted point y = x3 − x at x = 2 ?
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 ?
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 y = x3 − 2x2 − 2x at which the tangent lines are parallel to the line y = 2x− 3 ?
Find the points on the curve \[\frac{x^2}{9} + \frac{y^2}{16} = 1\] at which the tangent is parallel to x-axis ?
Find the equation of the tangent and the normal to the following curve at the indicated point \[\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1 \text { at } \left( x_0 , y_0 \right)\] ?
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 the tangent and the normal to the following curve at the indicated points x = asect, y = btant at t ?
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 x2 + y2 − 4x − 1 = 0 and x2 + y2 − 2y − 9 = 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 set of curve intersect orthogonally x3 − 3xy2 = −2 and 3x2y − y3 = 2 ?
Show that the curves 2x = y2 and 2xy = k cut at right angles, if k2 = 8 ?
Find the coordinates of the point on the curve y2 = 3 − 4x where tangent is parallel to the line 2x + y− 2 = 0 ?
Write the angle between the curves y = e−x and y = ex at their point of intersections ?
Write the slope of the normal to the curve \[y = \frac{1}{x}\] at the point \[\left( 3, \frac{1}{3} \right)\] ?
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 _____________ .
Any tangent to the curve y = 2x7 + 3x + 5 __________________ .
The two curves x3 – 3xy2 + 2 = 0 and 3x2y – y3 = 2 ______.
Prove that the curves xy = 4 and x2 + y2 = 8 touch each other.
Find the angle of intersection of the curves y = 4 – x2 and y = x2.
At what points on the curve x2 + y2 – 2x – 4y + 1 = 0, the tangents are parallel to the y-axis?
Show that the line `x/"a" + y/"b"` = 1, touches the curve y = b · e– x/a at the point where the curve intersects the axis of y
The slope of tangent to the curve x = t2 + 3t – 8, y = 2t2 – 2t – 5 at the point (2, –1) is ______.
At (0, 0) the curve y = x3 + x
The points on the curve `"x"^2/9 + "y"^2/16` = 1 at which the tangents are parallel to the y-axis are:
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 = 2x2 - x + 1 is parallel to the line y = 3x + 9 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 number of common tangents to the circles x2 + y2 – 4x – 6x – 12 = 0 and x2 + y2 + 6x + 18y + 26 = 0 is
Which of the following represent the slope of normal?
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 ______.
