Slope Form of Tangent: Hyperbola

Q. The locus of the foot of perpendicular from the centre upon any normal to the hyperbola $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$ is

1. $(x^2-y^2{)}^2(a^2{y}^2-b^2{x}^2)=(a^2+b^2{)}^2{x}^2{y}^2$
2. $(x^2+y^2{)}^2(a^2{y}^2-b^2{x}^2)=(a^2+b^2{)}^2{x}^2{y}^2$
3. $(x^2-y^2{)}^2(a^2{y}^2-b^2{x}^2)=(a^2-b^2{)}^2{x}^2{y}^2$
4. $(x^2+y^2{)}^2(a^2{y}^2-b^2{x}^2)=(a^2-b^2{)}^2{x}^2{y}^2$

Q.

The equations of the common tangents to the two hyperbolas

$\frac{x^2}{a^2}-\frac{y^2}{b^2}=1(a<b)$ and $\frac{y^2}{a^2}-\frac{x^2}{b^2}=1(a>b)$ is

1. $y=\pm x\pm \sqrt{a^2+b^2}$
2. $y=\pm x\pm \sqrt{a^3+b^3}$
3. $y=\pm x\pm \sqrt{a^2-b^2}$
4. $y=\pm x\pm \sqrt{a^3-b^3}$

Q. A tangent to the hyperbola $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$ cuts the ellipse $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$ in points $P$ & $Q$. The locus of the mid- point $PQ$ is

1. $(\frac{x^2}{a^2}-\frac{y^2}{b^2})={(\frac{x^2}{a^2}+\frac{y^2}{b^2})}^2$
2. $2(\frac{x^2}{a^2}-\frac{y^2}{b^2})={(\frac{x^2}{a^2}+\frac{y^2}{b^2})}^2$
3. $\frac{x^2}{a^2}+\frac{y^2}{b^2}={(\frac{x^2}{a^2}-\frac{y^2}{b^2})}^2$
4. $\frac{2x^2}{a^2}-\frac{y^2}{b^2}={(\frac{x^2}{a^2}+\frac{y^2}{b^2})}^2$

Q.

The circle $x^2+y^2-8x=0$ and hyperbola $\frac{x^2}{9}-\frac{y^2}{4}=1$ intersect at the points A and B.
Equation of a common tangent with positive slope to the circle as well as to the hyperbola is

1. $2x-5\sqrt{y}-20=0$

2. $2x-\sqrt{5}y+4=0$

3. $3x-4y+8=0$

4. $4x-3y+4=0$

Q.

If $y=mx+4$ is a tangent to both the parabolas, $y^2=4x$ and $x^2=2by$, then $b$ is equal to

1. $-64$

2. $128$

3. $-128$

4. $-32$

Q. The angle between lines joining the origin to the points of intersection of the line $\sqrt{3}x+y=2$ and the curve $y^2-x^2=4$ is

1. ${\tan }^{-1}\left(\frac{\sqrt{3}}{2}\right)$
2. $\frac{\pi }{2}$
3. ${\tan }^{-1}\left(\frac{2}{\sqrt{3}}\right)$
4. $\frac{\pi }{6}$

Q. The total number of tangents to the hyperbola $\frac{x^2}{9}-\frac{y^2}{4}=1$ that are perpendicular to the line $5x+2y-3=0$ is

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

Q. The equation of a tangent to the hyperbola $4x^2-5y^2=20$ parallel to the line $x-y=2$ is :

1. $x-y+1=0$
2. $x-y+7=0$
3. $x-y-3=0$
4. $x-y+9=0$

Q. The slope(s) of the common tangent(s) to the two hyperbolas $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$ and $\frac{y^2}{a^2}-\frac{x^2}{b^2}=1$ is/are

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

Q.

If in a $△ABC$, the altitudes from the vertices $A,B,C$ on the opposite side are in HP, then $\sin A,\sin B,$ and $\sin C$ are in

1. HP

2. Arithmetic – Geometric Progression

3. AP

4. GP

Q.

The points on the curve $y^3+3x^2=12y$ where the tangent is vertical (parallel to the$y-$axis), are

1. $\left(\pm \frac{4}{\sqrt{3}},-2\right)$

2. $\left(\pm \frac{\sqrt{11}}{3},1\right)$

3. $(0,0)$

4. $\left(\pm \frac{4}{\sqrt{3}},2\right)$

Q. The equation of the common tangent to $x^2=6y$ and $2x^2-4y^2=9$ can be

1. $x+y=1$
2. $x-y=1$
3. $x-y=\frac{3}{2}$
4. $x+y=\frac{9}{2}$

Q. If $P$ and $Q$ be two points on the hyperbola $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$, whose centre is $C$ such that $CP$ is perpnediuclar to $CQ,a<b$, then the value of $\frac{1}{C{P}^2}+\frac{1}{C{Q}^2}$ is

1. $\frac{b^2-a^2}{2ab}$
2. $\frac{1}{a^2}+\frac{1}{b^2}$
3. $\frac{1}{a^2}-\frac{1}{b^2}$
4. $\frac{2ab}{b^2-a^2}$

Q. The equations of the tangents to the hyperbola $x^2-4y^2=36$ which are perpendicular to the line $x-y+4=0$ are:

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

Q. If $2x-y+1=0$ is a tangent to the hyperbola $\frac{x^2}{a^2}-\frac{y^2}{16}=1$, then which of the following $CANNOT$ be sides of a right angled triangle?

1. $a,4,1$
2. $a,4,2$
3. $2a,8,1$
4. $2a,4,1$

Q. $PQ$ is the chord joining the points ${\theta }_1$ and ${\theta }_2$ on the hyperbola $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$. If ${\theta }_1-{\theta }_2=2\alpha ,\alpha \ne \frac{(2n+1)\pi }{2}$ . If $PQ$ touches the hyperbola $\frac{x^2}{A^2}-\frac{y^2}{b^2}=1$ then :

1. Length of transverse axis $=2a\sec \alpha \text{units}$
2. Length of transverse axis $=2a\cos \alpha \text{units}$
   
3. $e=\sqrt{1+\frac{b^2{\cos }^2\alpha }{a^2}}$
4. $e=\sqrt{1+\frac{b^2{\sec }^2\alpha }{a^2}}$

Q. The angle between the lines $\frac{x}{1}=\frac{y}{0}=\frac{z}{-1}$ and $\frac{x}{3}=\frac{y}{4}=\frac{z}{5}$ is
[Pb. CET 2002]

1. 
   

2. 
   

3. 
   

4. 
   

Q. Equation of one of the tangents passing through $(2,8)$ to the hyperbola $5x^2-y^2=5$ is

1. $3x-y-2=0$
2. $x+y+3=0$
3. $10x-8y-5=0$
4. $23x-3y-22=0$

Q.

Find the slope of the tangent to the curve $y=\frac{x-1}{x-2},x\ne 2$ at x=10.

Q.

The equation of the tangent from the point $\left(0,1\right)$ to the circle $x^2+y^2-2x-6y+6=0$, is

1. $3\left(x^2-y^2\right)+4xy-4x-6y+3=0$

2. $3y^2+4xy-4x-6y+3=0$

3. $3x^2+4xy-4x-6y+3=0$

4. $3\left(x^2+y^2\right)+4xy-4x-6y+3=0$

Q. Consider a function $f\left(x\right)=1+12|x|-3x^2$ defined on $[-2,5].$ The absolute difference between the global maximum and global minimum values of $f\left(x\right)$ is

Q.

If the curves $\frac{x^2}{a^2}+\frac{y^2}{12}=1$ and $y^3=8x$ intersect at right angles, then $a^2$ is equal to :

1. $16$

2. $12$

3. $8$

4. $4$

5. $2$

Q.

If the curve $y=a^x$ and $y=b^x$ intersect at angle $\alpha$ then, $\tan \alpha =$

1. $\frac{a-b}{1+ab}$

2. $\frac{\log a-\log b}{1+\log a\log b}$

3. $\frac{a+b}{1-ab}$

4. $\frac{\log a+\log b}{1-\log a\log b}$

Q. If the distance between two parallel tangents drawn to the hyperbola $\frac{x^2}{9}-\frac{y^2}{49}=1$ is $2$ unit, then which of the following is(are) correct?

1. Quadrilateral formed by the points of contact of the tangent is rectangle.
2. Slope of one pair of parallel tangent is $\frac{5}{2}$
3. Slope of one pair of parallel tangent is $-\frac{5}{2}$
4. area formed by the points of contact of the tangent is $180$ sq. units

Q. The equation of the common tangents to $x^2+4y^2=8$ and $y^2=4x$ can be

1. $2x+y-4=0$
2. $2x-y+4=0$
3. $x-2y+4=0$
4. $x+2y+4=0$

Q. For all real permissible values of $m,$ if the straight line $y=mx+\sqrt{9m^2-4}$ is tangent to a hyperbola, then equation of the hyperbola can be

1. $9x^2-4y^2=64$
2. $4x^2-9y^2=36$
3. $4x^2-9y^2=64$
4. $9x^2-4y^2=36$

Q. If the value's of $m$ for which the line $y=mx+2\sqrt{5}$ touches the hyperbola $16x^2-9y^2=144$ are roots of the equation $x^2-(a+b)x-4=0$, then the value of $a+b=$

Q. Tangent drawn from point $(c,d)$ to the hyperbola $\frac{x^2}{25}-\frac{y^2}{16}=1$ make angles $\alpha$ and $\beta$ with the $x-$ axis. If $\tan \alpha \tan \beta =1$, then the value of $c^2-d^2$

Q. If the line $y=mx+c$ touches $x^2-y^2=1$ and $y^2=4x,$ then $m^2$ is equal to

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

Q.

If the curves $2x^2+3y^2=6$ and $a{x}^2+4y^2=4$ intersect orthogonally then $a=$

1. $2$

2. $1$

3. $3$

4. $-3$