Question

the equation $\frac{x^{2}}{16}-\frac{(y - 5)^{2}}{36}=1$ represents a hyperbola when graphed in the xy - plane. if this graph is shifted 4 units down and 2 units right, what is the new equation of the hyperbola? a) $\frac{(x - 2)^{2}}{16}-\frac{(y - 1)^{2}}{36}=1$ b) $\frac{(x - 2)^{2}}{16}-\frac{(y - 9)^{2}}{36}=1$ c) $\frac{(x + 2)^{2}}{16}-\frac{(y - 1)^{2}}{36}=1$ d) $\frac{(x + 2)^{2}}{16}-\frac{(y - 9)^{2}}{36}=1$

Explanation:

Step1: Recall translation rules

For a right - shift of \(h\) units and a down - shift of \(k\) units, replace \(x\) with \(x - h\) and \(y\) with \(y + k\) in the original equation. Here \(h = 2\) and \(k=4\). The original equation is \(\frac{x^{2}}{16}-\frac{(y - 5)^{2}}{36}=1\).

Step2: Substitute the values of \(h\) and \(k\)

Replace \(x\) with \(x - 2\) and \(y\) with \(y+4\) in the original equation. We get \(\frac{(x - 2)^{2}}{16}-\frac{(y + 4-5)^{2}}{36}=1\).

Step3: Simplify the equation

Simplify \((y + 4 - 5)\) to \((y - 1)\). The new equation is \(\frac{(x - 2)^{2}}{16}-\frac{(y - 1)^{2}}{36}=1\).

Answer:

A. \(\frac{(x - 2)^{2}}{16}-\frac{(y - 1)^{2}}{36}=1\)