Point Form of Tangent: Hyperbola

Q. If the line $y=mx+7\sqrt{3}$ is normal to the hyperbola $\frac{x^2}{24}-\frac{y^2}{18}=1$, then a value of $m$ is

1. $\frac{2}{\sqrt{5}}$
2. $\frac{\sqrt{5}}{2}$
3. $\frac{3}{\sqrt{5}}$
4. $\frac{\sqrt{15}}{2}$

Q. If the eccentricity of the standard hyperbola whose transverse axis is $x-$axis and passing through the point $(4,6)$ is $2$, then the equation of the tangent to the hyperbola at $(4,6)$ is

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

Q. If the eccentricity of the standard hyperbola whose transverse axis is $x-$axis and passing through the point $(4,6)$ is $2$, then the equation of the tangent to the hyperbola at $(4,6)$ is

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

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. $P$ is a point on the hyperbola $\frac{x^2}{4}-\frac{y^2}{9}=1$, $N$ is the foot of perpendicular from $P$ on the transverse axis. The tangent to the hyperbola at $P$ meets the transverse axis at $T$. If $O$ is the centre of hyperbola, then the value of $OT\times ON$is

Q. The number of pairs of perpendicular tangents to the hyperbola $3x^2-6x-2y^2+4y-23=0$ is:

1. $0$
2. $1$
3. $2$
4. infinitely many

Q. If the tangent at a point $P(\alpha ,\beta )$ on the hyperbola $\frac{x^2}{25}-\frac{y^2}{16}=1$ cuts the circle $x^2+y^2=25$ at the point $Q(x_1,y_1)$ and $R(x_2,y_2),$ then $\frac{y_1{y}_2}{y_1+y_2}=$

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

Q. The shortest distance between the line $y=x$ and the hyperbola $\frac{x^2}{9}-\frac{y^2}{4}=1$ is

1. $\frac{3}{2}$ unit
2. $\frac{2}{3}$ unit
3. $\sqrt{\frac{5}{2}}$ unit
4. $9$ unit
   

Q. If the line $y=mx+7\sqrt{3}$ is normal to the hyperbola $\frac{x^2}{24}-\frac{y^2}{18}=1$, then a value of $m$ is

1. $\frac{2}{\sqrt{5}}$
2. $\frac{\sqrt{5}}{2}$
3. $\frac{3}{\sqrt{5}}$
4. $\frac{\sqrt{15}}{2}$

Q. If a normal to the hyperbola $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$ meets the axes at $M\&N$ and the lines $MP\&NP$ are drawn perpendicular to the axes meeting at $P$, then locus of $P$ is

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