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.

The angle between the tangents drawn from the point $(1,4)$ to the parabola $y^2=4x$ is

1. $0$

2. $\frac{\mathrm{\pi }}{6}$

3. $\frac{\mathrm{\pi }}{4}$

4. $\frac{\mathrm{\pi }}{3}$

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. The eccentricity of the hyperbola $3x^2-4y^2=-12$ is

1. 
2. 
3. 
4. 

Q.

The length of the normal at the point $t$ of the curve $x=a\left(t+\sin t\right)$, $y=a\left(1-\cos t\right)$ is

1. $a\sin \left(t\right)$

2. $2a{\sin }^3\left(\frac{t}{2}\right)\sec \left(\frac{\mathrm{t}}{2}\right)$

3. $2a\sin \left(\frac{t}{2}\right)\tan \left(\frac{\mathrm{t}}{2}\right)$

4. $2a\sin \left(\frac{t}{2}\right)$

Q.

If l₁, m₁, n₁ and l₂, m₂, n₂ are the direction cosines of two mutually perpendicular lines, show that the direction cosines of the line perpendicular to both of these are m₁n₂ − m₂n₁, n₁l₂ − n₂l₁, l₁m₂ ­− l₂m₁.

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.

The number of tangents to the curve $x^{\frac{3}{2}}+y^{\frac{3}{2}}=2a^{\frac{3}{2}},a>0$, which are equally inclined to the axes, is

1. $2$

2. $1$

3. $0$

4. $4$

Q.

The length of normal to the curve $x=a\left(\theta +\sin \left(\theta \right)\right),y=a\left(1-\cos \left(\theta \right)\right)$ at the point $\theta =\frac{\pi }{2}$ is

1. $2a$

2. $\frac{a}{2}$

3. $\sqrt{2}a$

4. $\frac{a}{\sqrt{2}}$

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

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

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 $\frac{\sqrt{109}}{2}$

Q. If $9+f"\left(x\right)+f^{\prime }\left(x\right)=x^2+f^2\left(x\right)$ be the differential equation of a curve and let P be the point of minima then the number of tangents which can be drawn from P to the circle $x^2+y^2=9$ is

1. 1
2. 0
3. either 1 or 2
4. 2

Q. Eccentricity of hyperbola $\frac{x^2}{k}+\frac{y^2}{k}=1(k<0)$ is :

1. 
2. 
3. 
4. 

Q. A tangent is drawn to the parabola $y^2=6x$ which is perpendicular to the line $2x+y=1$. Which of the following points does $\text{NOT}$ lie on it ?

1. $(-6,0)$
2. $(4,5)$
3. $(0,3)$
4. $(5,4)$

Q. A hyperbola has $y$-axis and $x$-axis as its conjugate axis and transverse axis respectively. If one of the points of intersection of $x$-axis with the hyperbola is $(4,0)$ and equation of one of the tangents is $x-y=\sqrt{7},$ then the equation of the hyperbola is

1. $\frac{x^2}{16}-\frac{y^2}{9}=1$
2. $\frac{x^2}{9}-\frac{y^2}{16}=1$
3. $\frac{x^2}{25}-\frac{y^2}{9}=1$
4. $\frac{x^2}{9}-\frac{y^2}{25}=1$

Q.

If a line, $y=mx+c$ is a tangent to the circle, ${(x-3)}^2+y^2=1$ and it is perpendicular to a line $l_1$, where $l_1$ is the tangent to the circle, $x^2+y^2=1$ at the point $\left(\frac{1}{\surd 2},\frac{1}{\surd 2}\right)$; then:

1. $c^2+7c+6=0$

2. $c^2-6c+7=0$

3. $c^2+6c+7=0$

4. $c^2-7c+6=0$

Q.

Prove that the line through the point (x₁, y₁) and parallel to the line Ax + By + C = 0 is A (x –x₁) + B (y – y₁) = 0.

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. $L_1$ 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 $\frac{\sqrt{109}}{2}$

Q. The eccentricity of a hyperbola whose transverse axis is along $x-$ axis and passes through the point $(6,4)$ is $\sqrt{\frac{5}{3}}$. The equation of tangent to this hyperbola at $(6,4)$ is

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

Q. The equation of common tangent to the curve $y^2=8x$ and $xy=-1$ is

1. $2y=x+2$
2. $y=x+2$
3. $3y=9x+2$
4. $y=x+8$

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

Q. The value of $p$ such that the length of the sub-tangent and sub-normal is equal for the curve $y=e^{px}+px$ at the point $(0,1)$ is

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

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 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. A normal to the hyperbola $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$ meets the transverse and conjugate axes in M and N and the lines MP and NP are drawn at right angle to the axes. The locus of P is

1. The parabola
2. The circle
3. The ellipse
4. The hyperbola

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 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. $(4,-\sqrt{6})$