Slope Form of Tangent: Hyperbola

Q.

What is the condition for a line $y=mx+c$ to be tangent to the hyperbola $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1.$

1. $c=\pm \sqrt{b^2-a^2{m}^2}$

2. $c=\pm \sqrt{a^2{m}^2-b^2}$

3. $c=\pm \sqrt{a^2-b^2{m}^2}$

4. $c=\pm \sqrt{b^2+a^2{m}^2}$

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. 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.

What is the equation of tangent to the hyperbola
$2x^2-3y^2=6$ which is parallel to the line $y=3x+4$, is

1. $y=3x-5$

2. $y=3x+5\text{and}y=3x-5$

3. None of the above

4. $y=3x+5$

Q. Columns $1,2$ and $3$ contain conics, equations of tangents to the conics and points of contact, respectively.
​​​​​​$\begin{array}{cccccc}& \text{Column}-1& & \text{Column}-2& & \text{Column}-3\\ \left(I\right)& x^2+y^2=a^2& \left(i\right)& my=m^{2}x+a& \left(P\right)& (\frac{a}{m^2},\frac{2a}{m})\\ \left(II\right)& x^2+a^2{y}^2=a^2& \left(ii\right)& y=mx+a\sqrt{m^2+1}& \left(Q\right)& (\frac{-ma}{\sqrt{m^2+1}},\frac{a}{\sqrt{m^2+1}})\\ \left(III\right)& y^2=4ax& \left(iii\right)& y=mx+\sqrt{a^2{m}^2-1}& \left(R\right)& (\frac{-a^{2}m}{\sqrt{a^2{m}^2+1}},\frac{1}{\sqrt{a^2{m}^2+1}})\\ \left(IV\right)& x^2-a^2{y}^2=a^2& \left(iv\right)& y=mx+\sqrt{a^2{m}^2+1}& \left(S\right)& (\frac{-a^{2}m}{\sqrt{a^2{m}^2-1}},\frac{-1}{\sqrt{a^2{m}^2-1}})\end{array}$

For $a=\sqrt{2}$, if a tangent is drawn to a suitable conic $\text{(Column 1)}$ at the point of contact $(-1,1)$, then which of the following options is the only $\text{CORRECT}$ combination for obtaining its equation?

1. $\left(I\right)\left(i\right)\left(P\right)$
2. $\left(I\right)\left(ii\right)\left(Q\right)$
3. $\left(II\right)\left(ii\right)\left(Q\right)$
4. $\left(III\right)\left(i\right)\left(P\right)$

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. 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.

What is the equation of tangent to the hyperbola
$2x^2-3y^2=6$ which is parallel to the line $y=3x+4$, is

1. $y=3x+5$

2. $y=3x-5$

3. $y=3x+5\text{and}y=3x-5$

4. None of the above

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. Common tangent equations to $9x^2-16y^2=144$ and $x^2+y^2=9$ are

1. $y=\pm \frac{3x}{\sqrt{7}}+\pm \frac{15}{\sqrt{7}}$
2. $y=\pm 3\sqrt{\frac{2}{7}}x\pm \frac{15}{\sqrt{7}}$
3. $y=\pm 2\sqrt{\frac{2}{7}}x\pm \frac{15}{\sqrt{7}}$
4. $y=\pm \frac{4x}{\sqrt{7}}\pm \frac{15}{\sqrt{7}}$

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. 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 equations to the common tangents to the two hyperbolas $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$ and $\frac{y^2}{a^2}-\frac{x^2}{b^2}=1$ are

1. $y=\pm x\pm \sqrt{(b^2-a^2)}$

2. $y=\pm x\pm \sqrt{(a^2-b^2)}$

3. $y=\pm x\pm (a^2-b^2)$

4. $y=\pm x\pm \sqrt{(a^2+b^2)}$

Q.

The line $x\cos \alpha +y\sin \alpha =p$ touches the hyperbola
$\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$ if

1. $a^2{\cos }^2\alpha -b^2{\sin }^2\alpha =p^2$

2. $a^2{\cos }^2\alpha -b^2{\sin }^2\alpha =p$
3. $a^2{\cos }^2\alpha -b^2{\cos }^2\alpha =2p$
4. $a^2{\cos }^2\alpha +b^2{\sin }^2\alpha =p$

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