Question
Download Solution PDFFind the equation of the hyperbola whose vertices are (±3, 0) and the foci are (±4, 0)
Answer (Detailed Solution Below)
Detailed Solution
Download Solution PDFConcept:
Equation of Hyperbola: \(\rm \frac{x^2}{a^2}- \frac{y^2}{b^2}=1\)
Eccentricity: \(\rm e= \rm\sqrt{1+ \frac{b^2}{a^2}} \)
Vertices = (±a, 0)
Focus = (±ae, 0)
Calculation:
The vertices of the given hyperbola are (±3, 0) which are in the form of (±a, 0)
∴ a = 3
Also, the foci are (±4, 0), which is in the form of (±ae, 0)
∴ ae = 4
⇒ e = 4/3 .....(∵ a = 3)
Now, we know, \(\rm e= \rm\sqrt{1+ \frac{b^2}{a^2}} \)
\(\rm \Rightarrow \frac43=\sqrt{1+\frac{b^2}{3^2}}\)
\(\rm \Rightarrow \frac{16}{9}={\frac{9+b^2}{9}}\) .....(squaring both sides)
\(\rm \Rightarrow 16={9+b^2}\)
⇒ b2 = 7
∴ Equation of hyperbola = \(\rm \frac{x^2}{9}- \frac{y^2}{7}=1\)
Hence, option (4) is correct.
Last updated on Jul 4, 2025
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