Advertisements
Advertisements
प्रश्न
Write the angle between the curves y = e−x and y = ex at their point of intersections ?
Advertisements
उत्तर
\[\text { Given }: \]
\[y = e^{- x} . . . \left( 1 \right)\]
\[y = e^x . . . \left( 2 \right)\]
\[\text { On substituting the value of y in (1), we get }\]
\[ e^{- x} = e^x \]
\[ \Rightarrow x = 0\]
\[\text { and }\]
\[y = 1 ...................[\text { From } (2)]\]
\[\text { On differentiating (1) w.r.t.x,we get }\]
\[\frac{dy}{dx} = - e^{- x} \]
\[ \Rightarrow m_1 = \left( \frac{dy}{dx} \right)_\left( 0, 1 \right) = - 1\]
\[\text { On differentiating (2) w.r.t.x,we get }\]
\[\frac{dy}{dx} = e^x \]
\[ \Rightarrow m_2 = \left( \frac{dy}{dx} \right)_\left( 0, 1 \right) = 1\]
\[ \because m_1 \times m_2 = - 1\]
Since the multiplication of the slopes is - 1.
So the slopes are perpendicular to each other.
\[ \therefore \text { Required angle } = \frac{\pi}{2}\]
APPEARS IN
संबंधित प्रश्न
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 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).
Find the equation of the normal to curve y2 = 4x at the point (1, 2).
The line y = mx + 1 is a tangent to the curve y2 = 4x if the value of m is
(A) 1
(B) 2
(C) 3
(D) 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 \[y = \sqrt{x^3} \text { at } x = 4\] ?
Find the slope of the tangent and the normal to the following curve at the indicted point x = a (θ − sin θ), y = a(1 − cos θ) at θ = −π/2 ?
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 slope of the tangent and the normal to the following curve at the indicted point y = (sin 2x + cot x + 2)2 at x = π/2 ?
At what point of the curve y = x2 does the tangent make an angle of 45° with the x-axis?
At what points on the curve y = 2x2 − x + 1 is the tangent parallel to the line y = 3x + 4?
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 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 \[x^\frac{2}{3} + y^\frac{2}{3}\] = 2 at (1, 1) ?
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 normal to the curve ay2 = x3 at the point (am2, am3) ?
The equation of the tangent at (2, 3) on the curve y2 = ax3 + b is y = 4x − 5. Find the values of a and b ?
Find the angle of intersection of the following curve y = x2 and x2 + y2 = 20 ?
Find the angle of intersection of the following curve 2y2 = x3 and y2 = 32x ?
Find the angle of intersection of the following curve x2 + y2 − 4x − 1 = 0 and x2 + y2 − 2y − 9 = 0 ?
Show that the following set of curve intersect orthogonally x3 − 3xy2 = −2 and 3x2y − y3 = 2 ?
Find the slope of the tangent to the curve x = t2 + 3t − 8, y = 2t2 − 2t − 5 at t = 2 ?
The equation of the normal to the curve y = x(2 − x) at the point (2, 0) is ________________ .
The curves y = aex and y = be−x cut orthogonally, if ___________ .
The angle of intersection of the parabolas y2 = 4 ax and x2 = 4ay at the origin is ____________ .
Any tangent to the curve y = 2x7 + 3x + 5 __________________ .
Find the condition for the curves `x^2/"a"^2 - y^2/"b"^2` = 1; xy = c2 to interest orthogonally.
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.
Find the angle of intersection of the curves y2 = 4ax and x2 = 4by.
The two curves x3 – 3xy2 + 2 = 0 and 3x2y – y3 = 2 ______.
Find an angle θ, 0 < θ < `pi/2`, which increases twice as fast as its sine.
The points on the curve `"x"^2/9 + "y"^2/16` = 1 at which the tangents are parallel to the y-axis are:
The tangent to the curve y = x2 + 3x will pass through the point (0, -9) if it is drawn at the point ____________.
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 line is y = x + 1 is a tangent to the curve y2 = 4x at the point.
The slope of the tangentto the curve `x= t^2 + 3t - 8, y = 2t^2 - 2t - 5` at the point `(2, -1)` is
Two vertical poles of heights, 20 m and 80 m stand apart on a horizontal plane. The height (in meters) of the point of intersection of the lines joining the top of each pole to the foot of the other, From this horizontal plane is ______.
