Corresponding Points : Ellipse

Q.

If two spheres of radii $r_1$ and $r_2$ cut orthogonally, then the radius of the common circle is

1. 
   

2. 
   

3. 
   

4. 
   

Q.

The area of the quadrilateral formed by the tangents at the endpoints of the latus rectum to the ellipse $\frac{{\mathrm{x}}^2}{9}+\frac{{\mathrm{y}}^2}{5}=1$, is

1. $\frac{27}{4}$sq. units

2. $9$sq. units

3. $\frac{27}{2}$sq. units

4. $27$sq. units

Q.

The inequality ${\sin }^{-1}\left(\sin 5\right)>x^2-4x$ holds if

1. $x=2–\sqrt{9–2\pi }$

2. $x=2+\sqrt{9–2\pi }$

3. $x\in \left(2–\sqrt{9–2\pi },2+\sqrt{9–2\pi }\right)$

4. $x>2+\sqrt{9–2\pi }$

Q. If $4x^2+y^2=1,$ then the maximum value of $12x^2-3y^2+16xy$ is

Q. If $x$ and $y$ are real numbers which satisfy the relation $x^2+9y^2-4x+6y+4=0,$ then the maximum value of $\frac{(4x-9y)}{2}$ is

Q.

The curve represented by $x=a(\sin h\theta +\cos h\theta )$, $y=b(-\sin h\theta +\cos h\theta )$ is

1. a hyperbola

2. a parabola

3. an ellipse

4. a circle

Q. If $x,y\in R$, satisfies the equation
$\frac{(x-4{)}^2}{4}+\frac{y^2}{9}=1,$ then the difference between the largest and smallest value of the expression $\frac{x^2}{4}+\frac{y^2}{9}$ is

Q. If the ratio of area of triangle inscribed in the ellipse $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$ to that of triangle formed by the corresponding points on the auxiliary circle is $\frac{1}{2}$, then the eccentricity of the ellipse is

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

Q. If $S$ and $S^{\prime }$ are the foci of the ellipse $\frac{x^2}{25}+\frac{y^2}{16}=1,$ and $P$ is any point on it then range of values of $SP\cdot S^{\prime }P$ is

1. $[9,16]$
2. $[16,25]$
3. $[1,10]$
4. $[9,25]$

Q. Let $A,B,C$ and $D$ be four points on the ellipse
$\frac{x^2}{a^2}+\frac{y^2}{b^2}=1,a>b,$ such that $AB$ and $CD$ cut its major axis at two distinct equidistant points from centre.
Let $\alpha ,\beta ,\gamma ,\delta$ represent eccentric angles of $A,B,C$ and $D$ respectively. Then
$\tan \frac{\alpha }{2}\tan \frac{\beta }{2}\tan \frac{\gamma }{2}\tan \frac{\delta }{2}=$

Q. Let the point $P(\alpha ,\beta )$ on the ellipse $4x^2+3y^2=12,$ in the first quadrant such that the area enclosed by the lines $y=x,y=\beta ,x=\alpha$ and the $x-$ axis is maximum, then

1. $P\equiv (\frac{3}{2},1)$
2. $P\equiv (\frac{\sqrt{3}}{2},\sqrt{3})$
3. eccentric angle of $P$ is $\frac{\pi }{3}$
4. eccentric angle of $P$ is $\frac{\pi }{6}$

Q. A point on the ellipse $\frac{x^2}{16}+\frac{y^2}{9}=1$ at a distance equal to the mean of the lengths of the semi major axis and semi minor axis from the centre is

1. $(\pm \frac{2\sqrt{91}}{7},\pm \frac{3\sqrt{105}}{14})$
2. $(\pm \frac{2\sqrt{91}}{7},\pm \frac{3\sqrt{105}}{7})$
3. $(\pm \frac{2\sqrt{105}}{14},\pm \frac{3\sqrt{91}}{14})$
4. $(\pm \frac{2\sqrt{105}}{7},\pm \frac{3\sqrt{91}}{14})$

Q. $P$ is a variable point on ellipse $9x^2+16y^2=144$ with foci $S$ and $S^{\prime }.$ If $K$ is the area of triangle $S{S}^{\prime }P,$ then the maximum value of $K^2$ is

Q.

The point of intersection of the tangents at the point P on the ellipse$\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$ and its corresponding point Q on the auxiliary circle, lies on the line

1. x = a/e

2. x = 0

3. y = 0

4. None of these

Q.

In a $\Delta$ ABC, let $\angle$ C = $\pi /2$ .If r is the inradius and R is the circumradius of the triangle, then 2 (r +R) is equal to

1. a + b

2. b + c

3. c + a

4. a + b + c

Q. A point on the ellipse $\frac{x^2}{16}+\frac{y^2}{9}=1$ at a distance equal to the mean of the lengths of the semi major axis and semi minor axis from the centre is

1. $(\pm \frac{2\sqrt{91}}{7},\pm \frac{3\sqrt{105}}{14})$
2. $(\pm \frac{2\sqrt{91}}{7},\pm \frac{3\sqrt{105}}{7})$
3. $(\pm \frac{2\sqrt{105}}{7},\pm \frac{3\sqrt{91}}{14})$
4. $(\pm \frac{2\sqrt{105}}{14},\pm \frac{3\sqrt{91}}{14})$

Q. The point of intersection of the tangents at the point $P$ on the ellipse $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$ and its corresponding point $Q$ on the auxiliary circle meet on the line :

1. $x=\frac{a}{e}$
2. $x=0$
3. $y=0$
4. None

Q. If $-\pi \le x\le \pi$, $-\pi \le y\le \pi$ and $\cos x+\cos y=2$ then $\cos (x-y)$

Q. If $OA$ is equally inclined to $OX,OY$ and $OZ$ and if $A$ is $\sqrt{3}$ units from the origin then $A$ is

1. $\left(3,3,3\right)$
2. $\left(-1,1,-1\right)$
3. $\left(-1,1,1\right)$
4. $\left(1,1,1\right)$

Q. The point $(4,1)$ undergoes the following transformation successively.
(i) Reflection about the line $y=x$
(ii) Translation through $a$ distance $2$ units along the positive direction of x-axis.
(iii) Rotation through an angle $\frac{\pi }{4}$ about the origin in the anticlockwise direction,
(iv) Reflection about $x=0$
The final position of the given point is

1. $(\frac{1}{\sqrt{2}},\frac{7}{2})$
2. $(\frac{1}{2},7\sqrt{2})$
3. $(\frac{1}{\sqrt{2}},\frac{7}{\sqrt{2}})$
4. $(\frac{1}{2},\frac{7}{2})$

Q. If the tangent at a point $(a\cos \theta ,b\sin \theta )$ on the ellipse $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$ meets the auxiliary circle in two points, the chord joining them subtends a right angle at the centre; then the eccentricity of the ellipse is given by

1. ${(1+{\sin }^2\theta )}^{-1/2}$
2. ${(1+{\cos }^2\theta )}^{-1/2}$
3. $1+{\cos }^2\theta$
4. $1+{\sin }^2\theta$

Q.

What is the corresponding point of $(acos\theta ,bsin\theta )$ of $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$ on its auxiliary cirlcle?

1. (a cosθ, a sin θ)

2. (b cosθ, b sin θ)

3. (a sin θ, b cos θ)

4. (a sin θ, a cos θ)

Q. If $x$ and $y$ are real numbers which satisfy the relation $x^2+9y^2-4x+6y+4=0,$ then the maximum value of $\frac{(4x-9y)}{2}$ is

Q. If $\overrightarrow{a}$ and $\overrightarrow{b}$ are unit vectors and $\theta$ is the angle between them, then $\left|\frac{\overrightarrow{a}-\overrightarrow{b}}{2}\right|$ is

1. $\sin \frac{\theta }{2}$
2. $\sin \theta$
3. $2\sin \theta$
4. $\sin 2\theta$

Q. If $x,y\in R$, satisfies the equation $\frac{(x-4{)}^2}{4}+\frac{y^2}{9}=1,$ then the difference between the largest and smallest value of the expression $\frac{x^2}{4}+\frac{y^2}{9}$ is

Q. If $S$ and $S^{\prime }$ are the foci of the ellipse $\frac{x^2}{25}+\frac{y^2}{16}=1,$ and $P$ is any point on it then range of values of $SP\cdot S^{\prime }P$ is

1. $[9,16]$
2. $[9,25]$
3. $[16,25]$
4. $[1,10]$

Q. $\underset{\theta \to 0}{lim}\left(\left[\frac{n\sin \theta }{\theta }\right]+\left[\frac{n\tan \theta }{\theta }\right]\right)$, where [.] represent greatest integer function and $n\in N$ is equal to

1. $2n$
2. $2n+1$
3. $2n-1$
4. does not exist

Q. If $0\le \theta \le \pi$ and ${81}^{{\sin }^2\theta }+{81}^{{\cos }^2\theta }=30$ then $\theta$ is

1. ${120}^{\circ }$
2. ${30}^{\circ }$
3. ${60}^{\circ }$
4. ${150}^{\circ }$

Q. The value of the integral ${\int }_0^{2\pi }\frac{sin2\theta }{a-bcos\theta }d\theta$ when a > b > 0, is

1. $\pi$
2. 1
3. $\pi /2$
4. 0

Q. Consider the geometric progression $S=1+2{\sin }^2\theta +4{\sin }^4\theta +8{\sin }^6\theta +.....$ up to infinite terms, where S is a finite number and $\theta \ne \frac{n\pi }{2}$ where n $\epsilon$ I.
Then Values of $\theta$ always lies in the interval ?

1. $(-\frac{\pi }{6},\frac{\pi }{6})$
2. $(0,\frac{\pi }{3})$
3. $(-\frac{\pi }{3},0)$
4. $(-\frac{\pi }{4},\frac{\pi }{4})-\left\{0\right\}$