If the line x + By + C = 0 is the normal to t | vedclass

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

Question

(A) Given the curve $x = a \sin^3 t$ and $y = b \cos^3 t$. Find the derivatives: $\frac{dx}{dt} = 3a \sin^2 t \cos t$ and $\frac{dy}{dt} = -3b \cos^2

Vedclass · Accepted Answer

$a$

Vedclass · Answer

(A) Given the curve $x = a \sin^3 t$ and $y = b \cos^3 t$. Find the derivatives: $\frac{dx}{dt} = 3a \sin^2 t \cos t$ and $\frac{dy}{dt} = -3b \cos^2 t \sin t$. The slope of the tangent is $\frac{dy}{dx} = \frac{dy/dt}{dx/dt} = \frac{-3b \cos^2 t \sin t}{3a \sin^2 t \cos t} = -\frac{b}{a} \cot t$. At $t = \frac{\pi}{2}$, $\frac{dy}{dx} = -\frac{b}{a} \cot(\frac{\pi}{2}) = 0$. The slope of the normal is $-\frac{1}{dy/dx}$, which is undefined (vertical line). However, at $t = \frac{\pi}{2}$, the point is $(x, y) = (a \sin^3(\frac{\pi}{2}), b \cos^3(\frac{\pi}{2})) = (a, 0)$. $A$ vertical line passing through $(a, 0)$ is $x = a$, or $x - a = 0$. Comparing $x - a = 0$ with $x + By + C = 0$, we get $B = 0$ and $C = -a$. Therefore, $B - C = 0 - (-a) = a$.

If the line $x + By + C = 0$ is the normal to the curve given by $x = a \sin^3 t$ and $y = b \cos^3 t$ (where $a, b \neq 0$) at the point $t = \frac{\pi}{2}$, then $B - C =$

Similar Questions

The coordinates of the point $P$ on the curve $x=a(\theta+\sin \theta), y=a(1-\cos \theta)$, where the tangent is inclined at an angle $\frac{\pi}{4}$ to the $x$-axis, are

For the curve ${y^n} = {a^{n - 1}}x$,the subnormal at any point is constant. The value of $n$ must be

The tangent to the curve $y = x^2 - 5x + 5$ which is parallel to the line $2y = 4x + 1$ also passes through the point

Let $C$ be a curve given by $y(x)=1+\sqrt{4x-3}$,$x>\frac{3}{4}$. If $P$ is a point on $C$ such that the tangent at $P$ has a slope of $\frac{2}{3}$,then a point through which the normal at $P$ passes is:

Find the equation of all lines having slope $-1$ that are tangents to the curve $y=\frac{1}{x-1}, x \neq 1$.

Vedclass Test Series

Exam Paper Generator

Online Exam Module