Let y=f(x) be any curve on the X-Y plane and | vedclass

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

Question

(A) Let $P(x, y)$ be a point on the curve $y=f(x)$ and $C(a, b)$ be a fixed point not on the curve.The distance $d$ between $P$ and $C$ is given by $d

Vedclass · Accepted Answer

$PC$ is perpendicular to the tangent at $P$

Vedclass · Answer

(A) Let $P(x, y)$ be a point on the curve $y=f(x)$ and $C(a, b)$ be a fixed point not on the curve.The distance $d$ between $P$ and $C$ is given by $d^2 = (x-a)^2 + (y-b)^2$.For $d$ to be a maximum or minimum, $d^2$ must also be a maximum or minimum.Differentiating $d^2$ with respect to $x$, we get $\frac{d}{dx}(d^2) = 2(x-a) + 2(y-b)\frac{dy}{dx} = 0$.This implies $(x-a) + (y-b)\frac{dy}{dx} = 0$, which can be written as $\frac{y-b}{x-a} = -\frac{1}{dy/dx}$.Here, $\frac{y-b}{x-a}$ is the slope of the line segment $PC$, and $\frac{dy}{dx}$ is the slope of the tangent at $P$.Since the product of their slopes is $-1$, the line segment $PC$ is perpendicular to the tangent at $P$.

Let $y=f(x)$ be any curve on the $X-Y$ plane and $P$ be a point on the curve. Let $C$ be a fixed point not on the curve. If the length $PC$ is either a maximum or a minimum, then:

Similar Questions

At which point on the curve $y^2 = x$ does the tangent make an angle of $45^{\circ}$ with the $x-$axis?

Let $P(h, k)$ be a point on the curve $y=x^{2}+7x+2$ nearest to the line $y=3x-3$. Then the equation of the normal to the curve at $P$ is:

The length of the normal to the curve $x = a(t + \sin t)$,$y = a(1 - \cos t)$ at point $t$ is:

Show that the tangents to the curve $y=7x^3+11$ at the points where $x=2$ and $x=-2$ are parallel.

The slope of the tangent of the curve $\left(\frac{x}{31}\right)^n + \left(\frac{y}{1209}\right)^n = 2$ at $(31, 1209)$ is:

Vedclass Test Series

Exam Paper Generator

Online Exam Module