Parametric Form of Tangent: Hyperbola

Q.

The locus of a point $P(\alpha ,\beta )$ moving under the condition that the line $y=\alpha x+\beta$ is a tangent to the hyperbola $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1.$

1. a parabola

2. a circle

3. an ellipse

4. a hyperbola

Q.

If tangents to the parabola $y^2=4x$ intersect the hyperbola at A & B, then find the locus of point of intersection of tangent at A and B.

1. 4y² + 81x = 0

2. y² + 81x = 0

3. 4y² + 16x = 0

4. y² + 16x = 0

Q. The value(s) of $m$, for which the line $y=mx+\frac{25\sqrt{3}}{3}$ , is a normal to the conic $\frac{x^2}{16}-\frac{y^2}{9}=1$ is/are

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

Q. If the normal at $P$ to the hyperbola $x^2-y^2=4$ meets the axes in $G$ and $g$ and $C$ is centre of the hyperbola, then

1. $PG=PC$
2. $Pg=PC$
3. $PG=Pg$
4. $Gg=2PC$

Q. If the line $2x+\sqrt{6}y=2$ touches the hyperbola $x^2-2y^2=4$ then the point of contact is

(IIT JEE 2004)

1. $(-2,\sqrt{6})$
2. $(-5,2\sqrt{6})$
3. $(\frac{1}{2},\frac{1}{\sqrt{6}})$
4. $(4,-\sqrt{6})$

Q.

Let $P(asec\theta ,btan\theta )$ and $Q(asec\varphi ,btan\varphi )$, where $\theta +\varphi =\frac{\pi }{2}$, be two points on the hyperbola $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$.
If (h, k) is the point of the intersection of the normals at P and Q, then k is equal to

1. $\frac{a^2+b^2}{a}$

2. $-\left(\frac{a^2+b^2}{a}\right)$

3. $\frac{a^2+b^2}{b}$

4. $-\left(\frac{a^2+b^2}{b}\right)$

Q. Line inclination with
$\begin{array}{cc}Positivex-direction& Slopeofline\\ 1.0^o& p.0\\ 2.{90}^o& q.-\sqrt{3}\\ 3.{120}^o& r.-\frac{1}{\sqrt{3}}\\ 4.{150}^o& s.Notdefined\end{array}$

1. 1 - p 2 - s 3 - q 4 - r

2. 1 - p 2 - s 3 - r 4 - q

3. 1 - s 2 - p 3 - q 4 - r

4. 1 - p 2 - q 3 - r 4 - s