Slope Form of Tangent: Hyperbola

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. Tangents are drawn from a point $P$ to the parabola $y^2=4ax$. If the chord of contact of the parabola is a tangent to the hyperbola $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$, then the locus of $P$ is

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

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 equation of tangents to the curve $9y^2-4x^2=1$ which is parallel to the line $3y=x+7$ is/are

1. $2x-6y=\sqrt{3}$
2. $2x-6y=-\sqrt{3}$
3. $16x-20y=25$
4. $16x-20y=-25$

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

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

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

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

Q. If the line $25x+12y-45=0$ meets the hyperbola $25x^2-9y^2=225$ only at point $(a,-\frac{5\lambda }{3}),$ then the value of $a+\lambda$ is

Q. The circle $x^2+y^2-8x=0$ and hyperbola $\frac{x^2}{9}-\frac{y^2}{4}=1$ intersects at the points $A$ and $B.$ Then

1. equation of common tangents will be $2x\pm \sqrt{5}y+4=0$
2. point of intersection of common tangents is $(-2,0)$
3. equation of common tangents will be $2x\pm \sqrt{5}y-4=0$
4. intersection point of common tangents will be $(2,0)$

Q. Equation of tangent(s) which passing through $(2,8)$ to the hyperbola $5x^2-y^2=5$ is (are)

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

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. 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 equation of tangents to the curve $9y^2-4x^2=1$ which is parallel to the line $3y=x+7$ is/are

1. $2x-6y=\sqrt{3}$
2. $2x-6y=-\sqrt{3}$
3. $16x-20y=25$
4. $16x-20y=-25$

Q. Let $x_1=\frac{1}{1\cdot 3}+\frac{1}{3\cdot 5}+\frac{1}{5\cdot 7}+\cdots$ upto $10$ terms,
$y_1=\text{Sum of roots of equation of}{x}^2-7x+10=0$. If $(x_1,y_1)$ lies inside the hyperbola $\frac{(21x{)}^2}{100}-\frac{y^2}{(7b{)}^2}=-1$, then number of integeral values for $b$ is

Q. If tangents are drawn to the hyperbola $4x^2-5y^2=20$ parallel to the line $x-y=2$, then

1. equation of tangent is $x-y=1$
2. point of contact will be $(-5,-4)$
3. point of contact will be $(5,4)$
4. equation of tangent is $x-y+1=0$

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. 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. 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. 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. $\left(III\right)\left(i\right)\left(P\right)$