# Definition of Ellipse

## Trending Questions

**Q.**

Let a vector $\begin{array}{l}\alpha \hat{i}+\beta \end{array}\hat{j}$ be obtained by rotating the vector $\begin{array}{l}\sqrt{3}\hat{i}+\hat{j}\end{array}$by an angle $45\xb0$ about the origin in counter clockwise direction in the first quadrant. Then the area of triangle having vertices $(\u0251,\beta ),(0,\beta )$ and $(0,0)$ is equal to:

$1$

$\frac{1}{2}$

$\sqrt{2}$

$2\sqrt{2}$

**Q.**If the latus rectum of an ellipse is equal to half of its minor axis, then its eccentricity is

- 32
- √32
- 23
- √23

**Q.**

A hyperbola having the transverse axis of length $\sqrt{2}$ has the same foci as that of the ellipse $3{x}^{2}+4{y}^{2}=12$, then this hyperbola does not pass through which of the following points -

$\left(\frac{\sqrt{3}}{2},\frac{1}{\sqrt{2}}\right)$

$\left(1,\frac{-1}{\sqrt{2}}\right)$

$\left(-\sqrt{\frac{3}{2}},1\right)$

$\left(\frac{1}{\sqrt{2}},0\right)$

**Q.**For the ellipse x2+4y2=9, which of the following option(s) is/are correct

- the eccentricity is 12
- the length of latus-rectum is 32
- one focus is at (3√3, 0)
- one directrix is x=−2√3

**Q.**

If three points A, B&C have position vectors (1, x, 3), (3, 4, 7) and (y, -2, -5) respectively.If they are collinear find the values of x&y.

**Q.**If the length of the latus rectum of an ellipse is 4 units and the distance between a focus and its nearest vertex on the major axis is 32 units, then its eccentricity is

- 23
- 19
- 12
- 13

**Q.**The slope of a line passing through P(2, 3) and intersecting the line, x+y=7 at a distance of 4 units from P, is:

- 1−√71+√7
- √5−1√5+1
- 1−√51+√5
- √7−1√7+1

**Q.**The equation of an ellipse, centred at origin and passing through the points (4, 3) and (−1, 4), is

- 7x2+15y2=247
- 49x2+225y2=105
- 7x2+15y2=105
- 7x2+15y2=147

**Q.**Let S and S′ be the two foci of the ellipse x2a2+y2b2=1. If the circle described on SS′ as diameter touches the ellipse in real points, then 6e2=

**Q.**The eccentricity of an ellipse 9x2+16y2=144 is

- √74
- 25
- √35
- √53

**Q.**From any point P lying in the first quadrant on the ellipse x225+y216=1, PN is drawn perpendicular to the major axis and produced at Q so that NQ equals to PS′, where S′ can be any foci. Then the locus of Q can be

- 3x−5y+25=0
- 3x+5y+25=0
- 3x+5y=25
- 3x−5y=25

**Q.**

If the distance between the foci of an ellipse is $6$ and the distance between its directrices is $12$, then the length of its latus rectum is

$4\sqrt{3}$

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

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

$3\sqrt{2}$

**Q.**

Why is the eccentricity of an ellipse between $0$ and $1$?

**Q.**If a straight line through the point P(λ, 2) where λ≠0, meets the ellipse x29+y24=1 at A and D and meets the coordinate axis at B and C such that PA⋅PD=PB⋅PC, then range of λ is

(correct answer + 2, wrong answer - 0.50)

- [12, ∞)
- (−∞, −12]∪[12, ∞)
- (−∞, −6]∪[6, ∞)
- [6, ∞)

**Q.**

C1 and C2 are circles of unit radius with centres at (0, 0) and (1, 0) respectively. C3 is a circle of unit radius passes through the centres of the circles C1 and C2 and have its centre above x-axis. Equation of the common tangent to C1 and C3 which does not pass through C2 is

**Q.**If the length of the major axis of the ellipse (5x−10)2+(5y+15)2=(3x−4y+7)24, is k units, then 3k=

**Q.**Consider two straight lines, each of which is tangent to both the circle x2+y2=12 and the parabola y2=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 √2, then which of the following statement(s) is (are) TRUE?

- For the ellipse, the eccentricity is 1√2
- For the ellipse, the eccentricity is 12
- For the ellipse, the length of the latus rectum is 1 unit
- For the ellipse, the length of the latus rectum is 12

**Q.**Equation of the ellipse with focus (3, −2),

eccentricity 34 and directrix 2x−y+3=0 is

- 44x2+36xy+71y2−588x+374y+959=0
- 44x2+36xy+71y2−374x−528y+756=0
- 44x2+36xy+71y2−135x−47y+859=0
- 44x2+36xy+71y2−125x−274y+659=0

**Q.**An underpass bridge is in the shape of a semi ellipse. It is 400 m long and has a maximum depth 10 m at the middle point. The depth of the bridge at a point which is at a distance of 80 m form one end is

**Q.**

Let A(a cos θ, b sin θ) is a variable point, S=(ap, 0) and S′ ≡ (−ap, 0) are two fixed points where p=√a2−b2a2. If the locus of Incentre of triangle ASS' is a conic then the eccentricity of the conic in terms of p is

- √p1+p
- √1−p1+p
- p2(1+p)
- √2p1+p

**Q.**A ladder 12 units long slides in a vertical plane with its ends in contact with a vertical wall and a horizontal floor along x−axis. Then the locus of a point on the ladder 4 units from its foot, is

- x264+y232=1
- x28+y24=1
- x264+y216=1
- x232+y232=1

**Q.**Let S1 and S2 are the unit circles with centres at C1(0, 0) and C2(1, 0) respectively. Let S3 is another circle of unit radius, passes through C1 and C2 and its centre is above the x -axis. If equation of common tangent to S1 and S3, which does not cut S2, is ax+by+2=0 then find the value of (a2−b).

**Q.**The number of the points where the function f(x)=min{|x|-1, |x-2|-1} is NOT derivable

**Q.**If P(x, y) is any point on the ellipse 16x2+25y2=400 andF1=(3, 0), F2=(−3, 0), then the value of PF1+PF2 is

- 2
- 5
- 10
- 20

**Q.**

Be the position of the point (-3, -2) with respect to the circle whose equation is x2+ y2-3x+2y-19=0

**Q.**the number of points where f(x) = |x^2 - 3x + 2| is non differentiable is

**Q.**Find the equation of ellipse whose focus is (1, −2), the directrix 3x−2y+5=0 and eccentricity equal to 12.

**Q.**Let P(x1, y1) and Q(x2, y2) where y1, y2<0, be the end points of the latus rectum of the ellipse x2+4y2=4. Then equation(s) of the parabola with latus rectum PQ is/are

- x2+2√3y=3+√3
- x2−2√3y=3+√3
- x2+2√3y=3−√3
- x2−2√3y=3−√3

**Q.**Find the equation of ellipse when: Focus is (0, 1), directrix is x+y=0 and e=12

**Q.**The eccentricity of the ellipse 25x2+16y2=400 is

- 35
- 13
- 15
- 25