Parametric Form of Tangent: Hyperbola

Q. Let $PQ:2x+y+6=0$ is a chord of the curve $x^2-4y^2=4.$ Coordinates of the point $R(\alpha ,\beta )$ that satisfy ${\alpha }^2+{\beta }^2-1\le 0;$ such that area of triangle $PQR$ is minimum, are given by :

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

Q.

The equation of the normal to the curve $y={(1+x)}^y+{\sin }^{-1}({\sin }^{2}x)$ at $x=0$ is

1. $x+y=1$

2. $x+y+1=0$

3. $2x–y+1=0$

4. $x+2y+2=0$

Q. If radii of director circles of $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$ and $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$ are $2r$ and $r$ respectively and $e_e$ and $e_h$ be the eccentricities of the ellipse and the hyperbola respectively then

1. $2e_h^2-e_e^2=6$
2. $e_e^2-4e_h^2=6$
3. $4e_h^2-e_e^2=6$
4. $e_h^2-2e_e^2=0$

Q. The area of triangle (in sq.units) formed by the tangents from point $(3,2)$ to hyperbola $x^2-9y^2=9$ and the chord of contact with respect to the point $(3,2)$ is

Q. If the tangents drawn to the hyperbola $4y^2=x^2+1$ intersect the co-ordinate axes at the distinct points $A$ and $B$, then the locus of the mid point of $AB$ is :

1. $x^2-4y^2+16x^2{y}^2=0$
2. $x^2-4y^2-16x^2{y}^2=0$
3. $4x^2-y^2+16x^2{y}^2=0$
4. $4x^2-y^2-16x^2{y}^2=0$

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.

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. The line $xcos\alpha +ysin\alpha =p$ touches the hyperbola $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$, if

1. $a^{2}co{s}^2\alpha -b^{2}si{n}^2\alpha =p^2$
   
2. $a^{2}co{s}^2\alpha -b^{2}si{n}^2\alpha =p$
   
3. $a^{2}co{s}^2\alpha +b^{2}si{n}^2\alpha =p^2$
   
   
4. $a^{2}co{s}^2\alpha +b^{2}si{n}^2\alpha =p$
   

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 hyperbola
2. a parabola
3. a circle
4. an ellipse

Q. Equation of tangent to the hyperbola $\frac{x^2}{a^2}-\frac{x^2}{b^2}=1$ at (asec$\theta$ , btan$\theta$) is $bxsec\theta -aytan\theta -ab=0$

1. True
2. false

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 hyperbola
2. a parabola
3. a circle
4. an ellipse

Q. A normal to the hyperbola, $4x^2-9y^2=36$ meets the co-ordinate axes $x$ and $y$ at $A$ and $B$, respectively. If the parallelogram $OAPB$ ($O$ being the origin) is formed, then the locus of $P$ is :

1. $4x^2-9y^2=121$
2. $4x^2+9y^2=121$
3. $9x^2-4y^2=169$
4. $9x^2+4y^2=169$

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 distance between two parallel tangents drawn to the hyperbola $\frac{x^2}{9}-\frac{y^2}{49}=1$ is $2$, then their slope is equal to

1. $\pm \frac{5}{2}$
2. $\pm \frac{4}{5}$
3. $\pm \frac{3}{2}$
4. None of these