TS EAMCET · Maths · Differential Equations
The differential equation of a family of hyperbolas whose axes are parallel to coordinate axes, centres lie on the line \(y=2 x\) and eccentricity is \(\sqrt{3}\) is
- A \((2 x-y) y_2+y_1^2-2 y_1=y_1^3+2\)
- B \((y-2 x) y_2+y_1^2+2 y_1=y_1^3+2\)
- C \((y-2 x) y_2-y_1^2+2 y_1=y_1^3-2\)
- D \((y+2 x) y_2+y_1^2+2 y_1=y_1^3-2\)
Answer & Solution
Correct Answer
(B) \((y-2 x) y_2+y_1^2+2 y_1=y_1^3+2\)
Step-by-step Solution
Detailed explanation
\(\text{Equation of hyperbola: } \frac{(x-h)^2}{a^2} - \frac{(y-k)^2}{b^2} = 1\) \(\text{Given } e=\sqrt{3} \Rightarrow b^2 = a^2(e^2-1) = a^2(3-1) = 2a^2\) \(\text{Center } (h,k) \text{ on } y=2x \Rightarrow k=2h\) \(\frac{(x-h)^2}{a^2} - \frac{(y-2h)^2}{2a^2} = 1\)…
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