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Ellipse from Focus (2, 3), Eccentricity 1/3 and Directrix Point (10, 11) | JEE Advanced

JEE Maths question with a full step-by-step solution.

Question
For an ellipse, one focus is (2,3)(2,3), the eccentricity is 13\dfrac13, and the corresponding directrix cuts the major axis at (10,11)(10,11). Which of the following statements are correct?
Athe vertices are (−2,−1)(-2,-1) and (4,5)(4,5)correct
Bthe major axis is 626\sqrt2correct
Cthe minor axis is 88correct
Dthe latus rectum is 3223\dfrac{32\sqrt2}{3}
Solution
Step 1: The major axis joins the focus S(2,3)S(2,3) to the directrix foot Z(10,11)Z(10,11), so
Z−S=(8,8),unit vector u^=12(1,1)Z - S = (8,8), \qquad \text{unit vector } \hat{u} = \frac{1}{\sqrt2}(1,1)
 and SZ=82+82=82SZ = \sqrt{8^{2}+8^{2}} = 8\sqrt2. Step 2: The focus is at distance aeae from the centre and the directrix at ae\dfrac{a}{e}, both on the same side, so
SZ=ae−ae=a(1−e2)eSZ = \frac{a}{e} - ae = \frac{a\left(1-e^{2}\right)}{e}
 Step 3: Putting e=13e = \dfrac13,
a(1−19)13=3a⋅89=8a3=82 ⟹ a=32\frac{a\left(1-\frac19\right)}{\frac13} = 3a\cdot\frac89 = \frac{8a}{3} = 8\sqrt2 \ \Longrightarrow\ a = 3\sqrt2
 Step 4: The nearer vertex is at distance a−ae=a(1−e)=32⋅23=22a - ae = a(1-e) = 3\sqrt2\cdot\tfrac23 = 2\sqrt2 from SS, towards ZZ:
A=(2,3)+22⋅12(1,1)=(2+2, 3+2)=(4,5)A = (2,3) + 2\sqrt2\cdot\frac{1}{\sqrt2}(1,1) = (2+2,\ 3+2) = (4,5)
 The farther vertex is at distance a+ae=a(1+e)=32⋅43=42a + ae = a(1+e) = 3\sqrt2\cdot\tfrac43 = 4\sqrt2 on the other side:
A′=(2,3)−42⋅12(1,1)=(2−4, 3−4)=(−2,−1).(1)A' = (2,3) - 4\sqrt2\cdot\frac{1}{\sqrt2}(1,1) = (2-4,\ 3-4) = (-2,-1) . \text{(1)}
 Step 5:
AA′=(4+2)2+(5+1)2=36+36=62=2a.(2)AA' = \sqrt{(4+2)^{2}+(5+1)^{2}} = \sqrt{36+36} = 6\sqrt2 = 2a . \text{(2)}
 Step 6:
2b=2a1−e2=62⋅89=62⋅223=8.(3)2b = 2a\sqrt{1-e^{2}} = 6\sqrt2\cdot\sqrt{\frac89} = 6\sqrt2\cdot\frac{2\sqrt2}{3} = 8 . \text{(3)}
 Step 7: From Step 6, b=4b = 4, so b2=16b^{2} = 16 and
ℓ=2b2a=3232=3226=1623\ell = \frac{2b^{2}}{a} = \frac{32}{3\sqrt2} = \frac{32\sqrt2}{6} = \frac{16\sqrt2}{3}
 which is half of the value printed in option (4), so (4) is false. Answer: (1), (2) and (3).
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