Ellipse Test

Result

Ellipse Test - 5

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Question 1
4 / -1

If maximum distance of any point on the ellipse x² + 2y² + 2xy = 1 from its centre be r, then r is equal to

A

B

C

D

Solution

Here centre of the ellipse is (0,0) Let P (r cos θ, r sinθ) be any point on the given ellipse then r² cos²θ + 2r² sin²θ + 2r² sin θ cos θ = 1
Question 2
4 / -1

The equation of the line passing through the centre and bisecting the chord 7x +y -1 = 0 of the ellipse

A
x = y

B
2x = y

C
x = 2y

D
x +y = 0

Solution

Let (h,k) be the midpoint of the chord 7 x +y -1 = 0

Represents same straight line

⇒ Equation of the line joining (0,0) and (h,k) is y - x = 0.
Question 3
4 / -1

Eccentricity of ellipse such that the line joining the foci subtends a right angle only at two points on ellipse, is

A
1/2

B
1/√2

C
1/√3

D
None of these

Solution
Question 4
4 / -1

The tangent at the point ‘a’ on the ellipse meets the auxiliary circle in two points which subtends a right angle at the centre, then the eccentricity ‘e’ of the ellipse is given by the equation :

A
e² (1 + cos² α) = 1

B
e² (cosec²α - 1) = 1

C
e² (1 + sin² α) = 1

D
e² (1 + tan² α) = 1

Solution

Equation of auxillary circle is x² + y² = a²
Let P is (acosa,bsina)
Equatio of AB is

bcosα.x+a sinαy = ab
To get combined equation of CA and CB, homogenize equation of circle with equation (i),

b²x² + b²y² - (bcosα.x+a sinα.y)² = 0
since ∠BCA = 90°
∴ coefficient of x² + coefficient of y² = 0
Question 5
4 / -1

The number of values of c such that the straight line y = 4x+ c touches the curve

A
0

B
1

C
2

D
Infinite

Solution

For given slope there exists two parallel tangents to ellipse. Hence, there are two values of c .
Question 6
4 / -1

If tangents are drawn to the ellipse x² + 2y² = 2, then the locus of the midpoint of the intercept made by the tangent between the coordinate axes is

A

B

C

D

Solution

For any tangent to ellipse

Using midpoint formula, we have
Question 7
4 / -1

An ellipse with major and minor axes, 6√3 and 6 respectively slides along the coordinate axes and always remains confined in the first quadrant. If the length of arc described by the centre of the ellipse is then the value of k is

A
6

B
3

C
9

D
4
Question 8
4 / -1

Consider the ellipse and the parabola y² = 2x. They intersect at P and Q in the first and fourth quadrants respectively. Tangents to the ellipse at P and Q intersect the x-axis at R and tangents to the parabola at P and Q intersect the x-axis at S. If the area of quadrilateral PQRS, is λ then

A
6

B
3

C
9

D
4

Solution

Area of quadrilateral
Question 9
4 / -1

The area of the quadrilateral formed by the tangents at the end points of latus rectum of the ellipse is k then k/9 is

A
6

B
3

C
9

D
4
Question 10
4 / -1

The length of the focal chord of the ellipse which is inclined to x – axis at an angle 45° is λ, then

A
2

B
3

C
4

D
5