Parametric Form of Tangent: Hyperbola

Q. A normal to the hyperbola $\frac{x^2}{4}-\frac{y^2}{1}=1$ has equal intercepts on positive $x$ and $y$ axes. If this normal touches the ellipse $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$, then the value of $3(a^2+b^2)$ is

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.

Let $a,b$and $\lambda$ be positive real numbers. Suppose $P$ is an end point of the latus rectum of the parabola $y^2=4\lambda$ , and suppose the ellipse $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$ passes through the point . If the tangents to the parabola and the ellipse at the point $P$ are perpendicular to each other, then the eccentricity of the ellipse is

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

2. $\frac{1}{2}$

3. $\frac{1}{3}$

4. $\frac{2}{5}$

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

Q.

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

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

2. $–\frac{[a^2+b^2]}{a}$

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

4. $–\frac{[a^2+b^2]}{b}$

Q. Equation of a common tangent with positive slope to the circle as well as to the hyperbola is

1. $3x-4y+8=0$
2. $2x-\sqrt{5}y+4=0$
3. $4x-3y+4=0$
4. $2x-\sqrt{5}y-20=0$

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 from any point on the asymptote a straight line be drawn perpendicular to the transverse axis, the product of the segments of this line, intercepted between the point & the curve is always equal to _____

1. square of the conjugate axis

2. square of the transverse axis

3. square of the semi-conjugate axis

4. square of the semi transverse axis

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.

The plane $x+2y-z=4$ cuts the sphere $x^2+y^2+z^2-x+z-2=0$in a circle of radius

1. $\sqrt{2}$

2. $2$

3. $1$

4. $3$

Q. If the tangent and normal to $x^2-y^2=a^2$ at a point cut off intercepts $a_1,a_2$ and $b_1,b_2$ on the $x$-axis and $y$-axis respectively, then $a_1{a}_2+b_1{b}_2$ is

1. $0$
2. $1$
3. $2$
4. Dependent on position of the point

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. If $A$ is the area formed by the positive $x$ axis, and the normal and tangent to the circle $x^2+y^2=4$
at $(1,\sqrt{3})$ then the value of $A/\sqrt{3}$ is

1. $2\sqrt{3}$
2. $2$
3. $\sqrt{3}$
4. $6$

Q. The distance of the point (2, 3) from the line 3x + 2y = 17, measured parallel to the line x – y = 4 is

1. 
2. 
3. 
4. 

Q. If a ray of light incident along the line $x=5$ gets reflected from the hyperbola $\frac{x^2}{16}-\frac{y^2}{9}=1$ at point $P$, then

1. equation reflected line is $9x+40y+45=0$
2. equation reflected line is $9x-40y+45=0$
3. area of $△PS{S}^{\prime }=22.5$ sq. unit
   $\text{(}S,S^{\prime }\text{are foci of hyperbola)}$
4. area of $△PS{S}^{\prime }=11.25$ sq. unit
   $\text{(}S,S^{\prime }\text{are foci of hyperbola)}$

Q.

The angle made by the tangent to the circle $x^2+y^2-8x+6y+20$ = 0 at (3, -1) with the positive direction of the x-axis is

1. 

2. 

3. 

4. 

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 equation of a straight line $3x+4y+12=0$ is transformed into normal form $x\cos \alpha +y\sin \alpha =p$, then the value of $\tan \alpha +\cot \alpha$ is equal to

1. $\frac{25}{12}$
2. $\frac{12}{25}$
3. $\frac{7}{12}$
4. $\frac{25}{7}$

Q. If the chords of contact of tangents from points $(x_1,y_1)\text{and}(x_2,y_2)$ to the hyperbola $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$ are at right angles such that $\frac{x_1{x}_2}{y_1{y}_2}=-\frac{a^m}{b^n}$ where $m,n$ are positve integers, then value of ${\left(\frac{m+n}{4}\right)}^{10}$ is

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

1. 
   
   
2. 
   
3. 
   
4. 
   

Q. If two normals on the parabola $(y-2{)}^2=-12(x+3)$ intersect each other at right angle then the chord joining their feet passes through a fixed point whose coordinate are

1. $(-2,2)$
2. $(-3,5)$
3. $(-6,2)$
4. $(-3,-1)$

Q. The altitudes from the vertices A, B and C of the triangle ABC meet it circumcircle at D, E

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.

Prove that the curves $x=y^2$ and xy=k cut at right angle if $8k^2=1$.

Q.

The equation of the tangent to the curve $x=2{\cos }^3\theta$ and$y=3{\sin }^3\theta$ at the point$\theta =\frac{\pi }{4}$ is

1. $2x+3y=3\sqrt{2}$

2. $2x-3y=3\sqrt{2}$

3. $3x+2y=3\sqrt{2}$

4. $3x-2y=3\sqrt{2}$

Q. Let $\tan \alpha ,\tan \beta ,\tan \gamma ;\alpha ,\beta ,\gamma \ne \frac{(2n-1)\pi }{2}$, $n\in N$ be the slopes of three line segment $OA,$ $OB$ and $OC,$ respectively, where $O$ is origin. If the circumcentre of $△ABC$ coincides with origin and its orthocentre lies on $y-$ axis, then the value of ${\left(\frac{\cos 3\alpha +\cos 3\beta +\cos 3\gamma }{\cos \alpha \cos \beta \cos \gamma }\right)}^2$ is equal to

Q. Let $\tan \alpha ,\tan \beta ,\tan \gamma ;\alpha ,\beta ,\gamma \ne \frac{(2n-1)\pi }{2}$, $n\in N$ be the slopes of three line segment $OA,$ $OB$ and $OC,$ respectively, where $O$ is origin. If the circumcentre of $△ABC$ coincides with origin and its orthocentre lies on $y-$ axis, then the value of ${\left(\frac{\cos 3\alpha +\cos 3\beta +\cos 3\gamma }{\cos \alpha \cos \beta \cos \gamma }\right)}^2$ is equal to

Q. The solution of the trigonometric equation
$\frac{{\cos }^2\theta }{{\cot }^2\theta -{\cos }^2\theta }=3,0^{\circ }<\theta <{90}^{\circ }$

1. $\theta =0^{\circ }$
2. $\theta ={60}^{\circ }$
3. $\theta ={30}^{\circ }$
4. $\theta ={90}^{\circ }$

Q. Let S be the circle in the xy-plane defined by the equation $x^2+y^2=4$. Let P be a point on the circle S with both coordinates being positive. Let the tangent to S at P intersect the coordinate axes at the points M and N. Then, the mid-point of the segment MN must lie on the curve.

1. $(x+y{)}^2=3xy$
2. $x^2+y^2=2xy$
3. $x^{2/3}+y^{2/3}=2^{4/3}$
4. $x^2+y^2=x^2{y}^2$

Q. In the interval $\left[-\frac{\pi }{2},\frac{\pi }{2}\right]$ the equation ${\log }_{\sin }\left(\cos 2\theta \right)=2$ has

1. no solution
2. a unique solution
3. two solutions
4. infinitely many solutions