Point Form of Tangent: Hyperbola

Q.

What is the point of contact between the hyperbola
$\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$ and
the tangent $y=mx\pm \sqrt{a^2{m}^2-b^2}$.

1. $[\pm \frac{a^{2}m}{\sqrt{a^2{m}^2-b^2}},\pm \frac{b^2}{\sqrt{a^2{m}^2-b^2}}]$

2. $[\pm \frac{b^2}{\sqrt{a^2{m}^2-b^2}},\pm \frac{a^{2}m}{\sqrt{a^2{m}^2-b^2}}]$

3. $[\pm \sqrt{a^2{m}^2-b^2},\pm \sqrt{a^2{m}^2+b^2}]$

4. $[\pm a^{2}m,\pm b^2]$

Q. If the line $2x+\sqrt{6}y=2$ touches the hyperbola
$x^2-2y^2=4$ then the point of contact is

1. $(-2,\sqrt{6})$
2. $(-5,2\sqrt{6})$
3. $(\frac{1}{2},\frac{1}{\sqrt{6}})$
4. $(4,-\sqrt{6})$

Q.

If the line $2x+\sqrt{6}y=2$ is tangent to the hyperbola $x^2-2y^2=4$ then the point of contact is.

1. 

2. 

3. 

4. 

Q. Consider th hyperbola $H:x^2-y^2=1$ and a circle s with centre $N(x_2,0).$ suppose that $H$ and $S$ touch each other at a point $P(x_1,y_1)$ with $x_1>and{y}_1>0$ the common tangent to $H$ and $S$ at $P$ intersects the X-axis at point M. If $(l,m)$ is the centroid of $\Delta PMN$; then the correct expression (s) is

1. 
   $\frac{dl}{d{x}_1}=1-\frac{1}{3{x_1}^2}for{x}_1>1$
2. $\frac{dm}{d{x}_1}=\frac{x_1}{3\sqrt{{x_1}^2-1}}for{x}_1>1$
3. $\frac{dl}{d{x}_1}=1+\frac{1}{3{x_1}^2}for{x}_1>1$
4. $\frac{dm}{d{y}_1}=\frac{1}{3}for{y}_1>0$

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 triangle?

1. $a,4,1$

2. $2a,4,1$

3. $a,4,2$

4. $2a,8,1$

Q. If a hyperbola passes through the point P $(\sqrt{2},\sqrt{3})$ and has foci $(\pm 2,0),$ then the tangent to this hyperbola at P also passes through the point

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

Q. Tangents are drawn to the hyperbola $\frac{x^2}{9}-\frac{y^2}{4}=1$
parallel to the straight line $2x-y=1$ The points of contacts of the tangents on the hyperbola are

1. $(\frac{9}{2\sqrt{2}},\frac{1}{\sqrt{2}})$
2. $(-\frac{9}{2\sqrt{2}},-\frac{1}{\sqrt{2}})$
3. $(3\sqrt{3},-2\sqrt{2})$
4. $(-3\sqrt{3},2\sqrt{2})$

Q. The point on the hyperbola $x^2-9y^2=9$, where the line 5x + 12y = 9 touches it, is .

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

Q. In a hyperbola portion of tangent intercepted between asymptotes is bisected at the point of contact. Consider a hyperbola whose centre is at origin. A line $x+y=2$ touches this hyperbola at $P(1,1)$ and intersects the asymtotes at $A$ and $B$ such that $AB=6\sqrt{2}$ units.

Equation of the tangent to the hyperbola at $(-1,\frac{7}{2})$ is

1. $5x+2y=2$
2. $3x+2y=4$
3. $3x+4y=11$
4. none of these

Q. $L_1$ is a tangent drawn to the curve $x^2-4y^2=16$ at $A(5,\frac{3}{2}).$ $L_2$ is another tangent parallel to $L_1$ which meets the curve at $B.$ $L_3$ and $L_4$ are normals to the curve at $A$ and $B$ respectively. Lines $L_1,L_2,L_3,L_4$ form a rectangle. Then

1. Equation of tangent at $B$ is $6y=5x-16$.
2. Equation of normal at $B$ is $12x+10y+75=0$.
3. Radius of largest circle inscribed in the rectangle is $\frac{32}{\sqrt{61}}$
4. Radius of the circle circumscribing the rectangle is $\frac{\sqrt{109}}{2}$