Equation of Normal at a Point (x,y) in Terms of f'(x)

Q. If the line aX + bY + c =0 is a normal to the curve xy =1. Then

1. a >0, b < 0
2. a > 0, b >0
3. a0
4. a < 0, b < 0

Q. reduce the equation x+y-root 2 =0 to the normal form x cos alpha +y sin alpha = p and hence find values of alpha and p

Q. ​the fix point through which the line (2cosθ+3sinθ)x + (3cosθ-5sinθ)y - (5cosθ-2sinθ)=0 passes for all values of θ is

Q. Find the perpendicular distance of the line joining the points (cos θ, sin θ) and (cos ϕ, sin ϕ) from the origin.

Q. The normal of the curve $x=d(cos\theta +\theta sin\theta )y=a(sin\theta -\theta cos\theta )$ at any $\theta$ is such that

[DCE 2000; AIEEE 2005]

1. It makes a constant angle with x-axis

2. It passes through the origin

3. It is at a constant distance from the origin

4. None of these

Q. If the normal of $y=f\left(x\right)$ at $(0,0)$ is given by $y-x=0,$ then
$\underset{x\to 0}{lim}\frac{x^2}{f\left(x^2\right)-20f\left(9x^2\right)+2f\left(99x^2\right)}$

1. equals $\frac{1}{19}$
2. equals $\frac{-1}{19}$
3. equals $\frac{1}{2}$
4. does not exist

Q. Let the line $y=mx$ and the ellipse $2x^2+y^2=1$ intersect at point $P$ in the first quadrant. If the normal to this ellipse at $P$ meets the co-ordinate axes at $(-\frac{1}{3\sqrt{2}},0)$ and $(0,\beta )$, then $\beta$ is equal to:

1. $\frac{2}{\sqrt{3}}$
2. $\frac{2}{3}$
3. $\frac{2\sqrt{2}}{3}$
4. $\frac{\sqrt{2}}{3}$

Q. How many numbers of pair ( x, y) satisfy the equation sin x + sin y = sin ( x + y ) and | x | + | y | = 1 , simultaneously . A> 1 B> 2 C> 4 D> 6

Q. reduce the equation x+ root 3 y -4=0 to the normal form x cos alpha +y sin alpha = p and hence find the values of alpha and p