Sign in
12thPass logo
New chatPYQ LibraryDoubtsRank report
Sign in to see Recents

Your guest activity stays on this device

Sign in to save progress →
Sign in

Ellipse question

2023 · 15 Apr · Shift 1 · Q39
Guest · filters and generic practice availableBrowsing as a guest · PYQ filters and generic practice are available. Sign in only for personalised features and saved progress.
  1. PYQ Library
  2. /JEE Main
  3. /Mathematics
  4. /Ellipse
  5. /2023 · 15 Apr · Shift 1 · Q39

Ellipse question

2023 · 15 Apr · Shift 1 · Q39

JEE MainMathematicsEllipseNumerical+4 / −1
Let an ellipse with centre (1,0)(1,0)(1,0) and latus rectum of length 12\frac{1}{2}21​ have its major axis along x\mathrm{x}x-axis. If its minor axis subtends an angle 60∘60^{\circ}60∘ at the foci, then the square of the sum of the lengths of its minor and major axes is equal to ‾\underline{\hspace{2cm}}​.
Numerical answer
View written solutionFree

Correct answer: 9

  1. Write the standard form of the ellipse

Since the ellipse has centre (1,0)(1,0)(1,0) and major axis along the xxx-axis, its equation can be written as

(x−1)2a2+y2b2=1,\frac{(x-1)^2}{a^2}+\frac{y^2}{b^2}=1,a2(x−1)2​+b2y2​=1,

where a>b>0a>b>0a>b>0.

Its foci are at

(1±c,0),c2=a2−b2.(1\pm c,0), \qquad c^2=a^2-b^2.(1±c,0),c2=a2−b2.
  1. Use the latus rectum length

For an ellipse, the length of the latus rectum is

2b2a.\frac{2b^2}{a}.a2b2​.

Given that this equals 12\frac1221​,

2b2a=12\frac{2b^2}{a}=\frac12a2b2​=21​

so

4b2=a.(1)4b^2=a. \tag{1}4b2=a.(1)
  1. Use the angle subtended by the minor axis at a focus

The endpoints of the minor axis are

(1,b)and(1,−b).(1,b) \quad \text{and} \quad (1,-b).(1,b)and(1,−b).

Take the right focus F=(1+c,0)F=(1+c,0)F=(1+c,0). The minor axis subtends angle 60∘60^\circ60∘ at this focus.

Vectors from FFF to the endpoints are

FP⃗=(−c,b),FQ⃗=(−c,−b).\vec{FP} = (-c,b), \qquad \vec{FQ}=(-c,-b).FP=(−c,b),FQ​=(−c,−b).

Then

cos⁡∠PFQ=FP⃗⋅FQ⃗∣FP⃗∣ ∣FQ⃗∣=c2−b2c2+b2.\cos \angle PFQ = \frac{\vec{FP}\cdot \vec{FQ}}{|\vec{FP}|\,|\vec{FQ}|} = \frac{c^2-b^2}{c^2+b^2}.cos∠PFQ=∣FP∣∣FQ​∣FP⋅FQ​​=c2+b2c2−b2​.

Given ∠PFQ=60∘\angle PFQ=60^\circ∠PFQ=60∘,

c2−b2c2+b2=12.\frac{c^2-b^2}{c^2+b^2}=\frac12.c2+b2c2−b2​=21​.

So,

2(c2−b2)=c2+b22(c^2-b^2)=c^2+b^22(c2−b2)=c2+b2 c2=3b2.(2)c^2=3b^2. \tag{2}c2=3b2.(2)

But for an ellipse,

c2=a2−b2.c^2=a^2-b^2.c2=a2−b2.

Hence from (2),

a2−b2=3b2a^2-b^2=3b^2a2−b2=3b2 a2=4b2a^2=4b^2a2=4b2 a=2b(a,b>0).(3)a=2b \quad (a,b>0). \tag{3}a=2b(a,b>0).(3)
  1. Solve for aaa and bbb

From (1),

a=4b2.a=4b^2.a=4b2.

Using (3),

2b=4b22b=4b^22b=4b2 b=12.b=\frac12.b=21​.

Then

a=2b=1.a=2b=1.a=2b=1.
  1. Find the sum of lengths of major and minor axes

Major axis length =2a=2=2a=2=2a=2.

Minor axis length =2b=1=2b=1=2b=1.

Their sum is

2a+2b=2+1=3.2a+2b=2+1=3.2a+2b=2+1=3.

Therefore, the required square is

(3)2=9.(3)^2=9.(3)2=9.
  1. Compare with stored answer

Derived answer = 999.

Stored correct answer = 999.

They agree.

PreviousNext

More from Ellipse

  • Let C be the largest circle centred at (2, 0) and inscribed in the ellipse 36x2​+16y2​=1. If (1, α) lies on C, then 10 α2 is equal to ​2023 · Numerical
  • Let the maximum area of the triangle that can be inscribed in the ellipse a2x2​+4y2​=1,a>2, having one of its vertices at one end of the major axis of the ellipse and one of its sides parallel…2022 · MCQ
  • If the ellipse a2x2​+b2y2​=1 meets the line 7x​+26​y​=1 on the x-axis and the line 7x​−26​y​=1 on the y-axis, then the eccentricity of the ellipse is :2022 · MCQ
  • The line y = x + 1 meets the ellipse 4x2​+2y2​=1 at two points P and Q. If r is the radius of the circle with PQ as diameter then (3r)2 is equal to :2022 · MCQ
  • The locus of the mid point of the line segment joining the point (4, 3) and the points on the ellipse x2+2y2=4 is an ellipse with eccentricity :2022 · MCQ
  • If the length of the latus rectum of the ellipse x2+4y2+2x+8y−λ=0 is 4 , and l is the length of its major axis, then λ+l is equal to ​.2022 · Numerical
  • Let the eccentricity of an ellipse a2x2​+b2y2​=1, a>b, be 41​. If this ellipse passes through the point (−452​​,3), then a2+b2 is…2022 · MCQ
  • Let a line L pass through the point of intersection of the lines bx+10y−8=0 and 2x−3y=0, b∈R−{34​}. If the line L also passes through the point (1,1) and touches the circle…2022 · MCQ