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

Q.

The normal to the curve at $P(x,y)$ meets the $x$-axis at $G$. If the distance of $G$ from the origin is twice the abscissa of $P$, then the curve is a

1. Ellipse

2. Parabola

3. Circle

4. Hyperbola

Q. If normal to the curve y = f(x) is parallel to x-axis, then correct statement is [RPET 2000]

1. $\frac{dy}{dx}=0$
2. $\frac{dy}{dx}=1$
3. $\frac{dx}{dy}=0$
4. None of these

Q. The distance, from the origin, of the normal to the curve, $x=2\cos t+2t\sin t,y=2\sin t–2t\cos t,\text{at}t=\frac{\pi }{4}$, is:

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

Q. Co-ordinates of a point on the curve y = x log x at which the normal is parallel to the line 2x - 2y = 3 are
[RPET 2000]

1. $(O_2,O)$
2. (e, e)
3. $(e^2,2e^2)$
4. $(e^{-2},2e^{-2})$

Q. Let the normal at a point $P$ on the curve $y^2-3x^2+y+10=0$ intersects the $y$-axis at $(0,\frac{3}{2})$. If $m$ is the slope of the tangent at $P$ to the curve, them $|m|$ is equal to