Terminologies Related to a Hyperbola

Q.

If $(0,\pm 4)$ and $(0,\pm 2)$ be the foci and vertices of a hyperbola, then its equation is

1. 

2. 

3. 

4. 

Q.

What is the equation of chord joining 2 points of a parabola given by parameters $t_{1}and{t}_2$.

1. 

2. 

3. 

4. 

Q.

The equation $\frac{x^2}{12-k}+\frac{y^2}{8-k}=1$ represents.

1. A hyperbola, if

2. A ellipse, if

3. A hyperbola, if

4. None of the above

Q.

The length of the chord intercepted by the circle $x^2+y^2=r^2$ on the line $\frac{x}{a}+\frac{y}{b}=1$

1. 

2. 

3. 

4. 

Q. Circle described on any focal chord of a parabola as diameter touches directrix.

1. True

2. False

Q.

What is the length of focal chord of a parabola $y^2=8xmakingangle{30}^{\circ }$ with x axis?

Q.

The distance between the directrices of the hyperbola $x^2-y^2=9$ is ___ .

1. $\frac{9}{\sqrt{2}}$

2. $\frac{3}{\sqrt{2}}$

3. $6\sqrt{2}$

4. $3\sqrt{2}$

Q.

The points of extrema of $f\left(x\right)={\int }_0^x\frac{\sin t}{t}dt$ in the domain $x>0$ are

1. $n\pi ;n=1,2,..$

2. $(2n+1)\frac{\mathrm{\pi }}{2};n=1,2,..$

3. $(4n+1)\frac{\mathrm{\pi }}{2};n=1,2,..$

4. $(2n+1)\frac{\mathrm{\pi }}{2};n=1,2,<...$

Q.

The length of the latusrectum of the parabola $2y^2+3y+4x-2=0$ is

1. 

2. 

3. 6

4. 2

Q.

Latus rectum of a hyperbola is 8 and its conjugate axis is equal to half the distance between the foci. What is the eccentricity of the hyperbola.

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

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

3. $\frac{2}{\sqrt{3}}$

4. None of these

Q.

The eccentricity of the hyperbola $4x^2-y^2-8x-8y-28$ is equal to ___ .

1. 2

2. $\sqrt{5}$

3. $\sqrt{7}$

4. 2

Q. If the chord joining the points $\alpha$ and $\beta$ on the ellipse $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$ subtends a right angle at (a, 0), then $\tan \left(\frac{\alpha }{2}\right)\tan \left(\frac{\beta }{2}\right)=$

1. 
2. 
3. 
4. 

Q.

Equation of the hyperbola passing through (2, 1) and having the distance between the Directrices is $\frac{4}{\sqrt{3}}$ is

1. 5x²-2y² =18

2. 3x²-2y² =10

3. 2x²-y² =7

4. 3x²-5y² =7

Q.

The equation of a circle of radius 1 touching the circles $x^2+y^2-2|x|=0$ is

1. 

2. 

3. 

4. 

Q. If S is the focus and PQ is a focal chord of the parabola $y^2=4ax$ then SP the semilatusrectum, and SQ are in

1. AP

2. GP

3. HP

4. AGP

Q.

The equation of the transverse and conjugate axis of the hyperbola $16x^2-y^2+64x+4y+44=0$ are

1. 

2. 

3. 

4. None of these

Q.

Find the equation of chord joining the two parametric points of the $P\left(t_1\right)\&P\left(t_2\right)$ of the hyperbola $xy=c^2$.

1. \)

2. \)

3. \)

4. \)

Q.

For the hyperbola $\frac{x^2}{co{s}^2\alpha }-\frac{y^2}{si{n}^2\alpha }=1,$ when $\alpha$ changes, remains constant.

1. abscissa of vertices
2. abscissa of foci
3. eccentricity
4. directrix

Q. If $S_1,S_2,S_3$ are the sum of first $n$ natural numbers, their squares and their cubes, respectively, show that ${9S}_2^2=S_3(1+{8S}_1).$

Q. If $x-iy=\sqrt{\frac{a-ib}{c-id}}$ prove that ${(x^2+y^2)}^2=\frac{a^2+b^2}{c^2+d^2}$

Q.

Examine if the two circles$x^2+y^2-2x-4y=0and{x}^2+y^2-8y-4=0$ _________.

1. Touch each other internally

2. Touch each other externally

3. One circle lies completely outside the other circle

4. Intersect each other at two points

5. One circle lies completely inside the other circle

Q. The point (1, 2) is one extremity of focal chord of parabola $y^2=4x$. The length of this focal chord is

1. 2
2. 4
3. 6
4. none of these

Q.

For the hyperbola $\frac{x^2}{co{s}^2\alpha }-\frac{y^2}{si{n}^2\alpha }=1$

Which one of the following remain constant with change of $\alpha$ ?

1. Abscissa of vertices

2. abscissa of foci

3. Eccentricity

4. directrix

Q. If $a+ib=\frac{{(x+i)}^2}{{2x}^2+1}$, prove that $a^2+b^2=\frac{{(x^2+1)}^2}{{(2x^2+1)}^2}$.

Q.

Equation of the hyperbola with length of the latusrectum 4 and e = 3 is

1. 3x²-15y² =1

2. 16x²-2y² =1

3. 8x²-y² =1

4. 2x²-16y² =1

Q. The normal at the point $(b{t}_1^2,2b{t}_1)$ on a parabola meets the parabola again in the point $(b{t}_2^2,2b{t}_2)$, then

1. $t_2=-t_1-\frac{2}{t_1}$
2. $t_2=-t_1+\frac{2}{t_1}$
3. $t_2=t_1-\frac{2}{t_1}$
4. $t_2=t_1+\frac{2}{t_1}$

Q. The closest distance of the origin from a curve given as $a\overline{z}+\overline{a}z+a\overline{a}=0$ (a is a complex number) is

1. 
2. 
3. 1
4. 

Q.

The equation of the Latus Rectum of the hyperbola $\frac{{(x-4)}^2}{16}-\frac{{(y-3)}^2}{20}=1$ are____.

1. $x=1\pm 5.$

2. $y=2\pm 6.$

3. $x=4\pm 6.$

4. $y=3\pm 5.$

Q.

If the eccentricity of the hyperbola $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$ is 5/4 and 2x+3y-6 = 0 is a focal chord of the hyperbola, then the length of the transverse axis is equal to ___ .

1. $\frac{24}{5}$

2. $\frac{24}{7}$

3. $\frac{12}{5}$

4. $\frac{12}{7}$

Q. $a_n={(-1)}^{n-1}{5}^{n+1}$