The value of (a) and (b) for the hyperbola are [tex]\sqrt 3[/tex] and 2
The equation of the hyperbola is given as:
[tex]\frac{y^2}{b^2} - \frac{x^2}{a^2} = 1[/tex]
The curve of the graph passes through points (0,2) and (-4,3).
So, we have:
[tex]\frac{2^2}{b^2} - \frac{0^2}{a^2} = 1[/tex] and [tex]\frac{(-4)^2}{b^2} - \frac{3^2}{a^2} = 1[/tex]
[tex]\frac{2^2}{b^2} - \frac{0^2}{a^2} = 1[/tex] becomes
[tex]\frac{4}{b^2} = 1[/tex]
Cross multiply
[tex]b^2 = 4[/tex]
Take positive square root of both sides
[tex]b = 2[/tex]
[tex]\frac{(-4)^2}{b^2} - \frac{3^2}{a^2} = 1[/tex] becomes
[tex]\frac{(-4)^2}{2^2} - \frac{3^2}{a^2} = 1[/tex]
[tex]\frac{(-4)^2}{4} - \frac{9}{a^2} = 1[/tex]
[tex]4 - \frac{9}{a^2} = 1[/tex]
Collect like terms
[tex]- \frac{9}{a^2} = -3[/tex]
Cross multiply
[tex]-3a^2 = -9[/tex]
Divide both sides by -3
[tex]a^2 = 3[/tex]
Take square roots of both sides
[tex]a = \sqrt 3[/tex]
Hence, the value of (a) and (b) for the hyperbola are [tex]\sqrt 3[/tex] and 2
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