Position of a Line W.R.T Hyperbola

Q. The line $2x+y=1$ is a tangent to the hyperbola $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1.$ If this line passes through the point of intersection of the directrix and $x-$axis, then eccentricity of hyperbola is

Q. The perpendicular from the origin to the tangent at any point on the curve is equal to the abscissa of the point of contact. If equation of tangent to the curve at (1, 3) is ax + by + 5 = 0, then value of a + b is equal to?

Q.

Consider the hyperbola xy = 4 and a line 2x + y = 4. O is the centre of the hyperbola. Tangent at any point P on the hyperbola intersects the coordinate axes at A and B.

Locus of circumcentre of $\Delta OAB$ is ___

1. An ellipse with $e=\frac{1}{\sqrt{2}}$

2. A hyperbola with $e=\sqrt{2}$

3. A circle

4. A parabola

Q.

Prove that the curves x = y² and xy = k cut at right angles if 8k² = 1. [Hint: Two curves intersect at right angle if the tangents to the curves at the point of intersection are perpendicular to each other.]

Q.

If $y=mx$ be one of the bisectors of the angle between the lines $a{x}^2-2hxy+b{y}^2=0$, then

1. $h\left(1+m^2\right)+m\left(a-b\right)=0$

2. $h\left(1-m^2\right)+m\left(a+b\right)=0$

3. $h\left(1-m^2\right)+m\left(a-b\right)=0$

4. $h\left(1+m^2\right)+m\left(a+b\right)=0$

Q. If the distances from the origin of the centres of three circles $x^2+y^2+2{\lambda }_{i}x-c^2=0$ $(i=1,2,3)$ are in G.P., then the lengths of tangents drawn to them from any point on the circle $x^2+y^2=c^2$ are in

1. A.P.
2. G.P.
3. H.P.
4. None of these

Q. $\mathrm{The}\mathrm{value}\left(s\right)\mathrm{of}\lambda \mathrm{for}\mathrm{which}\mathrm{the}\mathrm{line\; y=2x+\lambda }$touches the hyperbola $16x^2-9y^2=144$is/are:

1. $\sqrt{5}$
2. -$\sqrt{5}$
3. $2\sqrt{5}$
4. $-2\sqrt{5}$

Q. If $(a\sec \theta ,b\tan \theta )$ and $(a\sec \varphi ,b\tan \varphi )$ are the ends of a focal chord of $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$, the value of $\tan \frac{\theta }{2}\cdot \tan \frac{\varphi }{2}$ can be equal to :

1. $\frac{e-1}{1+e}$
2. $\frac{1-e}{1+e}$
3. $\frac{1+e}{1-e}$
4. $\frac{e+1}{e-1}$

Q. Coordinates of the focus of the parabola $\sqrt{\frac{x}{a}}+\sqrt{\frac{y}{b}}=1$ is

1. $(\frac{a{b}^2}{a^2+b^2},\frac{a^{2}b}{a^2+b^2})$
2. $(\frac{ab}{a+b},\frac{ab}{a+b})$
3. $(\frac{a^{2}b}{a+b},\frac{a{b}^2}{a+b})$
4. $(a,b)$

Q. 5. if tangent and normal to the curve y=2sinx+sin2x are drawn at a point p=pi/3then find the area of quadrilateral formed by the tangent normal and oordinate, x axis.

Q. If the maximum value of $(x+y{)}^2\text{is}\lambda$ and $P(x,y)$ satisfies $x^2+y^2=1$, then the number of tangents that can drawn from $(\lambda ,0)$ to the hyperbola $(x-2{)}^2-y^2=1$ is

1. $0$
2. $1$
3. $2$
4. $3$

Q. The sides $AC$ and $AB$ of a $△ABC$ touches the conjugate hyperbola of the hyperbola $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$ at $C$ and $B.$ If the vertex $A$ lies on the ellipse $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1,$ then the side $BC$ always touches

1. $\text{an ellipse}$
2. $\text{a parabola}$
3. $\text{a circle}$
4. $\text{a hyperbola}$

Q. $\mathrm{The}\mathrm{value}\left(s\right)\mathrm{of}\lambda \mathrm{for}\mathrm{which}\mathrm{the}\mathrm{line\; y=2x+\lambda }$touches the hyperbola $16x^2-9y^2=144$is/are:

1. $\sqrt{5}$
2. -$\sqrt{5}$
3. $2\sqrt{5}$
4. $-2\sqrt{5}$

Q.

The condition that the line $\frac{x}{p}+\frac{y}{p}=1$ o be tangent to $\frac{x^z}{a^z}+\frac{y^z}{b^z}=1$ is

1. $\frac{a^z}{p^z}-\frac{b^z}{q^z}$ = 1

2. $\frac{a^z}{p^z}-\frac{b^z}{q^z}$ = 1

3. $\frac{a^z}{q^z}-\frac{b^z}{p^z}$ = 1

4. $\frac{a^z}{q^z}-\frac{b^z}{p^z}$ = 1

Q. The equation of the normal to the hyperbola $x^2-4y^2=5$ at (3, -1) is

1. 4x – 3y + 5 = 0
2. 4x + 4y + 15 = 0
3. 4x – 3y – 15 = 0
4. 4x + 3y – 15 = 0

Q. Let PQ be a focal chord of the parabola $y^2=4x$. The tangents to the parabola at P and Q meet at a point on
y = 2x + 1.

If PQ subtends an angle $\theta$ at the vertex of $y^2=4x$, then $tan\theta$ =

1. $-\frac{2\sqrt{5}}{3}$
2. $\frac{2\sqrt{5}}{3}$
3. $\frac{2\sqrt{7}}{3}$
4. $-\frac{2\sqrt{7}}{3}$

Q. In a $△ABC,$ let $a,b$ and $c$ denote the lengths of sides opposite to vertices $A,B$ and $C$ respectively. If $a=7,b=3,c=5$ and
$\frac{ab}{(\tan \frac{B}{2}+\tan \frac{C}{2})(\tan \frac{A}{2}+\tan \frac{C}{2})}+\frac{bc}{(\tan \frac{A}{2}+\tan \frac{C}{2})(\tan \frac{A}{2}+\tan \frac{B}{2})}+\frac{ac}{(\tan \frac{B}{2}+\tan \frac{C}{2})(\tan \frac{A}{2}+\tan \frac{B}{2})}$
equals $k,$ then the value of $\sqrt{k}+\frac{1}{2}$ is

Q. Given an isosceles triangle with equal sides of length $b$, base angle $\alpha <\frac{\pi }{4}$ and $R,r$ are the radii of circumcircle and incircle and $O,I$ are the centres of circumcircle and incircle respectively. Then:

1. $R=\frac{1}{2}b\text{cosec}\alpha$
2. $\Delta =2b^2\sin 2\alpha$
3. $r=\frac{b\sin 2\alpha }{2(1+\cos \alpha )}$
4. $OI=\left|\frac{b\cos \left(\frac{3\alpha }{2}\right)}{2\sin \alpha \cos \left(\frac{\alpha }{2}\right)}\right|$

Q. 4.FIND THE VALUES OF P AND Q IF THE SLOPE OF THE TANGENT TO THE CURVE XY+PX+QY=2 AT (1, 1) IS 2.

Q. If $A\left(\theta \right)$ and $B\left(\varphi \right)$ are the parametric ends of a chord of hyperbola $\frac{x^2}{16}-\frac{y^2}{9}=1$ which passes through $(4,0)$, then the value of $\tan \frac{\theta }{2}\cdot \tan \frac{\varphi }{2}$is

1. $0$
2. $2$
3. $1$
4. $-1$

Q.

If from any point on the circle $x^2+y^2=a^2$, tangents are drawn to the circle $x^2+y^2=b^2(a>b)$ then the angle between tangents is

1. 
2. 
3. 
4. 

Q. Let $AB$ be the focal chord of a parabola $y=x^2-2x+2$ with point $A\equiv (3,5)$ . Then the slope of normal at point $B$ would be :

1. $6$
2. $\frac{2}{3}$
3. $4$
4. $-0.25$

Q.

The slope of a line is double of the slope of another line. If tangent of the angle between them is, find the slopes of he lines.

Q. The number of integral values of exhaustive set of $a^2$ such that there exists a tangent to the ellipse $x^2+a^2{y}^2=a^2$ and the portion of the tangent intercepted by the hyperbola $a^2{x}^2-y^2=1$ subtends a right angle at the centre of the curves is

Q. Let $A=[-1,1],B=[-1,1],C=[0,\infty )$. Define $R_1=\left\{\right(x,y)\in A\times B:x^2+y^2=1\}$ and $R_2=\left\{\right(x,y)\in A\times C:x^2+y^2=1\}$. Then choose the correct option:

1. $R_1$ is a function and range is $[-1,1]$
2. $R_2$ is a function and range is $[-1,1]$
3. $R_2$ is a function and range is $[0,1]$
4. $R_1$ is a function and range is $[0,1]$

Q.

Length of the straight line $x-3y=1$ intercepted by the hyperbola $x^2-4y^2=1$ is

1. 

2. 

3. 

4. 

Q.

Length of the straight line $x-3y=1$ intercepted by the hyperbola $x^2-4y^2=1$ is

1. $\sqrt{10}$

2. $\frac{6}{5}$

3. $\frac{1}{\sqrt{10}}$

4. $\frac{6}{5}\sqrt{10}$

Q.

Find the slope of the tangent to the curve, x ≠ 2 at x = 10.

Q. If $y=mx+c$ touches the parabola $y^2=4a(x+a)$, then

1. $c=am+\frac{a}{m}$
2. $c=\frac{a}{m}$
3. $c=a+\frac{a}{m}$
4. None of these

Q. Find the slope of the tangent to the curve x = t² + 3t − 8, y = 2t² − 2t − 5 at t = 2.