Diameter : Hyperbola

Q. The locus of the centroid of the triangle formed by any point $P$ on the hyperbola $16x^2-9y^2+32x+36y-164=0$, and its foci is

1. $16x^2-9y^2+32x+36y-36=0$
2. $9x^2-16y^2+36x+32y-144=0$
3. $9x^2-16y^2+36x+32y-36=0$
4. $16x^2-9y^2+32x+36y-144=0$

Q. If two distinct tangents can be drawn from the point $(\alpha ,2)$ on different branches of the hyperbola
$\frac{x^2}{9}-\frac{y^2}{16}=1$, then

1. $|\alpha |<\frac{3}{2}$
2. $|\alpha |>\frac{2}{3}$
3. $|\alpha |>3$
4. $\alpha =1$

Q. The diameter of \(16x^2- 9 y^2 = 144\) which is conjugate to x = 2y, is

1. 
   
2. 
   
3. 
4. 

Q.

The radius of the circle with the polar equation $r^2-8r(\sqrt{3}\cos \theta +\sin \theta )+15=0$ is

1. $8$

2. $7$

3. $6$

4. $5$

Q. If $\theta$ is the acute angle between the tangents drawn from $(1,4)$ to the parabola $y^2=12x,$ then the value of $\sin \theta +2\cos \theta$ is equal to

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

Q. The circle $x^2+y^2-8x=0$ and hyperbola $\frac{x^2}{9}-\frac{y^2}{4}=1$ intersect at point $A$ and $B$. The equation of circle with $AB$ as diameter is:

1. $x^2+y^2-12x+24=0$
2. $x^2+y^2+12x+24=0$
3. $x^2+y^2+29x-12=0$
4. $x^2+y^2-24x-12=0$

Q. If the tangent to the curve $y=e^x$ at a point $(c,e^c)$ and the normal to the parabola $y^2=4x$ at the point $(1,2)$ intersect at the same point on the $x$-axis, then the value of $c$ is

Q. The circle $x^2+y^2-8x=0$ and hyperbola $\frac{x^2}{9}-\frac{y^2}{4}=1$ intersect at point $A$ and $B$. The equation of circle with $AB$ as diameter is:

1. $x^2+y^2+12x+24=0$
2. $x^2+y^2-12x+24=0$
3. $x^2+y^2+29x-12=0$
4. $x^2+y^2-24x-12=0$

Q. $L:x^2+2gxy+y^2=0$ represents the equation of pair of straight lines passing through the origin. If $L$ makes an acute angle of $\theta$ with the straight line $y=x$, then

1. $\sec 2\theta =-g$
2. $\cos \theta =\sqrt{\frac{g-1}{2g}}$
3. $\tan \theta =\sqrt{\frac{g+1}{g-1}}$
4. $\cot \theta =\sqrt{\frac{g+1}{g-1}}$

Q. Evaluate:
$\frac{{\left(0.43\right)}^3+{\left(1.47\right)}^3+{\left(1.1\right)}^3-3\times 0.43\times 1.47\times 1.1}{{\left(0.43\right)}^2+{\left(1.47\right)}^2+{\left(1.1\right)}^2-0.43\times 1.43-0.43\times 1.1-1.47\times 1.1}$

1. $1.90$
2. $2.87$
3. $3$
4. $3.47$

Q. The diameter of \(16x^2- 9 y^2 = 144\) which is conjugate to x = 2y, is

1. $y=\frac{16x}{9}$
   
2. $y=\frac{32x}{9}$
   
3. $x=\frac{16y}{9}$
4. $x-\frac{32y}{9}$

Q. If two distinct tangents can be drawn from the point $(\alpha ,2)$ on different branches of the hyperbola
$\frac{x^2}{9}-\frac{y^2}{16}=1$, then

1. $|\alpha |<\frac{3}{2}$
2. $|\alpha |>\frac{2}{3}$
3. $|\alpha |>3$
4. $\alpha =1$