Length of Normal

Q.

Length of normal is defined as the length between the point where the normal is drawn on the curve and the point where the normal intersects the y -axis

Q. If $\overrightarrow{a}\mathrm{and}\overrightarrow{b}$are two vectors such that $\left|\overrightarrow{a}.\overrightarrow{b}\right|=\left|\overrightarrow{a}\times \overrightarrow{b}\right|,$ write the angle between $\overrightarrow{a}\mathrm{and}\overrightarrow{b}.$

Q.

Length of normal is defined as the length of the normal between the point where the normal is drawn on the curve and the point where the normal intersects the y -axis

1. True

2. False

Q. The length of normal to the curve $x=a\left(\theta +\sin \theta \right),y=a\left(1-\cos \theta \right)\mathrm{at}\theta =\frac{\pi }{2}$ is .

1. $2\left(a\right|$
2. $\sqrt{2}\left(a\right|$
3. $\frac{\left(}{a}$
4. $\frac{\left(}{a}$

Q. The length of normal to the curve $x=a\left(\theta +\sin \theta \right),y=a\left(1-\cos \theta \right)\mathrm{at}\theta =\frac{\pi }{2}$ is .

1. $2\left|a\right|$
2. $\sqrt{2}\left|a\right|$
3. $\frac{\left|}{a}$
4. $\frac{\left|}{a}$

Q. The length of normal to the curve x = $a(\theta +sin\theta )$ at the point $\theta =\pi /2$ is
[RPET 1999]

1. 2a
2. a/2
3. $\sqrt{2}a$
4. $a/\sqrt{2}$

Q. The length of the normal at point ‘t’ of the curve x = a(t + sin t), y = a (1 - cos t) is
[RPET 2001]

1. a sin t
2. $2asi{n}^3\left(t/2\right)sec\left(t/2\right)$
3. 2a sin(t / 2) tan(t/2)
4. 2a sin (t / 2)

Q. The length of normal to the curve x = $a(\theta +sin\theta )$ at the point $\theta =\pi /2$ is
[RPET 1999]

1. 2a
2. a/2
3. $\sqrt{2}a$
4. $a/\sqrt{2}$

Q. The length of normal to the curve x = $a(\theta +sin\theta )$ at the point $\theta =\pi /2$ is
[RPET 1999]

1. a/2
2. $\sqrt{2}a$
3. $a/\sqrt{2}$
4. 2a

Q. The length of the normal at point ‘t’ of the curve x = a(t + sin t), y = a (1 - cos t) is
[RPET 2001]

1. $2asi{n}^3\left(t/2\right)sec\left(t/2\right)$
2. 2a sin(t / 2) tan(t/2)
3. 2a sin (t / 2)
4. a sin t

Q. If x = cos θ, y = sin³ θ, prove that $y\frac{d^{2}y}{d{x}^2}+{\left(\frac{dy}{dx}\right)}^2=3{\sin }^2\mathrm{\theta }\left(5{\cos }^2\mathrm{\theta }-1\right)$.

Q. Ans fast

Q14. If x = a, cos $\mathrm{\theta }$ + b sin ​$\mathrm{\theta }$ and y = a sin $\mathrm{\theta }$ – b cos $\mathrm{\theta }$, show that $y^2\frac{d^{2}y}{d{x}^2}-x\frac{dy}{dx}+y=0$.

Q. The angle between the curves y² = x and x² = y at (1, 1) is

(a)
(b)
(c) 90°
(d) 45°

Q. The length of normal to the curve $x=a\left(\theta +\sin \theta \right),y=a\left(1-\cos \theta \right)\mathrm{at}\theta =\frac{\pi }{2}$ is .

1. $2\left(a\right|$
2. $\sqrt{2}\left(a\right|$
3. $\frac{\left(}{a}$
4. $\frac{\left(}{a}$