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MCQEasyJEE 2023Conic Sections (Parabola, Ellipse, Hyperbola)

JEE Mathematics 2023 Question with Solution

Let HH be the hyperbola, whose foci are (1±2,0)(1 \pm \sqrt{2}, 0) and eccentricity is 2\sqrt{2}. Then the length of its latus rectum is:

  • A

    22

  • B

    33

  • C

    52\frac{5}{2}

  • D

    32\frac{3}{2}

Answer

Correct answer:A

Step-by-step solution

Standard Method

Given: The foci are (1±2,0)(1 \pm \sqrt{2}, 0) and the eccentricity is 2\sqrt{2}.

Find: The length of the latus rectum of the hyperbola.

The center is (1,0)(1,0), so the distance of each focus from the center is

c=2c = \sqrt{2}

Using the relation

e=cae = \frac{c}{a}

we get

2=2a\sqrt{2} = \frac{\sqrt{2}}{a}

Hence,

a=1a = 1

For a hyperbola,

c2=a2+b2c^2 = a^2 + b^2

So,

b2=c2−a2=2−1=1b^2 = c^2 - a^2 = 2 - 1 = 1

Now the length of the latus rectum is

L.R.=2b2aL.R. = \frac{2b^2}{a}

Substituting a=1a=1 and b2=1b^2=1,

L.R.=2⋅11=2L.R. = \frac{2 \cdot 1}{1} = 2

Therefore, the length of the latus rectum is 22. The correct option is A.

The solution shows one place where the option label is printed as C, but the worked value is clearly 22. Since option A is 22, the correct answer is A.

Using rectangular hyperbola observation

Given: e=2e=\sqrt{2} and the foci are (1±2,0)(1 \pm \sqrt{2},0).

Find: The latus rectum length.

Since the center is the midpoint of the foci, it is (1,0)(1,0) and

c=2c = \sqrt{2}

From

e=cae = \frac{c}{a}

we obtain

a=ce=22=1a = \frac{c}{e} = \frac{\sqrt{2}}{\sqrt{2}} = 1

Also, e=2e=\sqrt{2} indicates a rectangular hyperbola, for which

a=ba=b

Thus b=1b=1 and hence

L.R.=2b2a=2⋅121=2L.R. = \frac{2b^2}{a} = \frac{2 \cdot 1^2}{1} = 2

Therefore, the correct option is A.

Common mistakes

  • Using the distance between the two foci as cc instead of the distance from the center to one focus. Here 2c=222c = 2\sqrt{2}, so c=2c=\sqrt{2}. Always take cc as center-to-focus distance.

  • Applying the ellipse relation c2=a2−b2c^2 = a^2 - b^2 instead of the hyperbola relation c2=a2+b2c^2 = a^2 + b^2. For hyperbola, this sign is positive.

  • Using the wrong latus rectum formula. For the standard hyperbola x2a2−y2b2=1\frac{x^2}{a^2}-\frac{y^2}{b^2}=1, the latus rectum length is 2b2a\frac{2b^2}{a}, not 2a2b\frac{2a^2}{b}.

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