Exams Joint Entrance Examination Mathematics Hyperbola Question 51 Question 51 of 106 Medium MCQ
Let the eccentricity of the hyperbola H : x 2 a 2 − y 2 b 2 = 1 H:\dfrac{x^2}{a^2}-\dfrac{y^2}{b^2}=1 H : a 2 x 2 − b 2 y 2 = 1 satisfy 1 + b 2 a 2 = 5 2 1+\dfrac{b^2}{a^2}=\dfrac52 1 + a 2 b 2 = 2 5 , and let the length of its latus rectum be 6 2 6\sqrt{2} 6 2 . If y = 2 x + c y=2x+c y = 2 x + c is a tangent to H H H , then the value of c 2 c^2 c 2 is equal to:
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b 2 a 2 = 3 2 \dfrac{b^2}{a^2}=\dfrac32 a 2 b 2 = 2 3 . Latus rectum: 2 b 2 a = 6 2 ⇒ 2 ⋅ 3 2 a = 6 2 ⇒ a = 2 2 , a 2 = 8 , b 2 = 12 \dfrac{2b^2}a=6\sqrt{2}\Rightarrow2\cdot\dfrac32a=6\sqrt{2}\Rightarrow a=2\sqrt{2},\ a^2=8,\ b^2=12 a 2 b 2 = 6 2 ⇒ 2 ⋅ 2 3 a = 6 2 ⇒ a = 2 2 , a 2 = 8 , b 2 = 12 .
Tangent condition: c 2 = a 2 m 2 − b 2 = 8 ( 4 ) − 12 = 20. c^2=a^2m^2-b^2=8(4)-12=20. c 2 = a 2 m 2 − b 2 = 8 ( 4 ) − 12 = 20. Answer the question to unlock