If the tangent of the curve y=e^{x} at a poin | vedclass

Name: Exam Paper Generator | JEE NEET Paper Generator | Vedclass
Brand: Vedclass
Rating: 5 (35 reviews)

Question

(B) For the curve $y=e^{x}$,the slope of the tangent at $(c, e^{c})$ is given by $\frac{dy}{dx} = e^{x} \implies m_{t} = e^{c}$.The equation of the ta

Vedclass · Accepted Answer

$4$

Vedclass · Answer

(B) For the curve $y=e^{x}$,the slope of the tangent at $(c, e^{c})$ is given by $\frac{dy}{dx} = e^{x} \implies m_{t} = e^{c}$.The equation of the tangent at $(c, e^{c})$ is $y - e^{c} = e^{c}(x - c)$.To find the intersection with the $x$-axis,set $y=0$: $-e^{c} = e^{c}(x - c) \implies -1 = x - c \implies x = c - 1$.For the parabola $y^{2}=4x$,differentiating gives $2y \frac{dy}{dx} = 4 \implies \frac{dy}{dx} = \frac{2}{y}$.At the point $(1,2)$,the slope of the tangent is $m = \frac{2}{2} = 1$.The slope of the normal is $m_{n} = -\frac{1}{m} = -1$.The equation of the normal at $(1,2)$ is $y - 2 = -1(x - 1) \implies y - 2 = -x + 1 \implies x + y = 3$.To find the intersection with the $x$-axis,set $y=0$: $x = 3$.Since the points of intersection on the $x$-axis are the same,we equate the $x$-coordinates: $c - 1 = 3 \implies c = 4$.

If the tangent of the curve $y=e^{x}$ at a point $(c, e^{c})$ and the normal to the parabola $y^{2}=4x$ at the point $(1,2)$ intersect at the same point on the $x$-axis,then the value of $c$ is

Similar Questions

The area of the triangle formed by the coordinate axes and a tangent to the curve $xy = a^2$ at the point $(x_1, y_1)$ is . . . . . . sq. units (where $a, x_1$,and $y_1$ are non-zero).

The points on the curve $y^2 = x + \sin x$ at which the normal is parallel to the $Y$-axis lie on

The point$(s)$ at which the tangents to the curve $y = x^3 - 3x^2 - 7x + 6$ cut off a line segment on the positive semi-axis $OX$ that is half the length of the segment cut off on the negative semi-axis $OY$ are given by:

The equation of the tangent to the curve $y=\sqrt{9-2x^2}$ at the point where the ordinate and abscissa are equal is:

If $x=t^2$ and $y=2t$ are parametric equations of a curve,then the equation of the normal to the curve at $t=2$ is

Vedclass Test Series

Exam Paper Generator

Online Exam Module