Definition of Ellipse

Q. Let $P(x_1,y_1)$ and $Q(x_2,y_2)$ where $y_1,y_2<0,$ be the end points of the latus rectum of the ellipse $x^2+4y^2=4.$ Then equation(s) of the parabola with latus rectum $PQ$ is/are

1. $x^2+2\sqrt{3}y=3+\sqrt{3}$
2. $x^2-2\sqrt{3}y=3+\sqrt{3}$
3. $x^2+2\sqrt{3}y=3-\sqrt{3}$
4. $x^2-2\sqrt{3}y=3-\sqrt{3}$

Q. A straight line through A(6, 8) meets the curve $2x^2+y^2=2$ at B and C. P is a point on BC such that AB, AP, AC are in H.P, then the minimum distance of the origin from the locus of ‘P’ is

1. 
2. 
3. 
4. 

Q.

The equation of $2x^2+3y^2-8x-18y+35=k$ represents

1. 

2. 

3. 

4. 

Q.

Find the area of the shaded region of the circle of radius $\mathrm{a}$ when the chord is $\mathrm{h}$ units $(0<\mathrm{h}<\mathrm{a})$ from the center of the circle (see figure).

Q. The number of integer points exactly in the interior of the triangle with vertices (0, 0) (0, 10) (10, 0) is k, then total number of combinations of x and y is a. 32 c. 36 b. 34 d. None of these

Q.

the number of points in(-infinity to +infinity) for which x^2-xsinx-cosx=0is1) 12) 43)64)0

Q.

Equation of the ellipse with vertices (-4, 3) (8, 3) and e=$\frac{5}{6}$ is

1. 

2. 

3. 

4. 

Q. Let $5x^2+7y^2=140$, then Position of $(4,-3)$ relative to the ellipse is

1. Inside the ellipse.
2. Outside the ellipse.
3. On the ellipse.
4. None of the above.

Q. { If the line }y=\sqrt3x cut the curve }x^3+y^3+3xy+5x^2+3y^2+4x+5y-1=0 at the points }A, B, C, } then OA. OB. OC is equal to (where O is origin)

Q. If the length of the major axis of the ellipse $(5x-10{)}^2+(5y+15{)}^2=\frac{(3x-4y+7{)}^2}{4},$ is $k$ units, then $3k=$

Q. Consider two straight lines, each of which is tangent to both the circle $x^2+y^2=\frac{1}{2}$ and the parabola $y^2=4x$. Let these lines intersect at the point $Q$. Consider the ellipse whose center is at the origin $O(0,0)$ and whose semi-major axis is $OQ$. If the length of the minor axis of this ellipse is $\sqrt{2}$, then which of the following statement(s) is (are) TRUE?

1. For the ellipse, the eccentricity is $\frac{1}{\sqrt{2}}$
2. For the ellipse, the eccentricity is $\frac{1}{2}$
3. For the ellipse, the length of the latus rectum is $1$ unit
4. For the ellipse, the length of the latus rectum is $\frac{1}{2}$

Q. The equation of reflection of the ellipse $\frac{(x-4{)}^2}{16}+\frac{(y-3{)}^2}{9}=1$ about the line $x-y-2=0$ is
​​​​​​​(correct answer + 1, wrong answer - 0.25)

1. $\frac{(x-2{)}^2}{16}+\frac{(y-5{)}^2}{9}=1$
2. $\frac{(x-2{)}^2}{9}+\frac{(y-5{)}^2}{16}=1$
3. $\frac{(x-5{)}^2}{16}+\frac{(y-2{)}^2}{9}=1$
4. $\frac{(x-5{)}^2}{9}+\frac{(y-2{)}^2}{16}=1$

Q. The argument of $\frac{1-i}{1+i}$ is
(a) $-\frac{\mathrm{\pi }}{2}$
(b) $\frac{\mathrm{\pi }}{2}$
(c) $\frac{3\mathrm{\pi }}{2}$
(d) $\frac{5\mathrm{\pi }}{2}$