Question

47. Which equation can be graphically represented as follows?

• A. \(8x^2-15y^2=169\)
• B. \(9x^2-16y^2=144\)
• C. \(|(x-8)(y-15)|=12\)
• D. \(|(x-9)(y-16)|=13\)

Hint:

Look at where the curve crosses the x-axis, then plug that point into each option to see which equation is satisfied.

Answer 1

The Correct Option is B

Step 1: Read off the key feature of the graph.
The picture shows two mirror-image curves opening left and right (a hyperbola opening along the x-axis), symmetric about both axes. Both branches touch the x-axis between the gridlines marked 2 and 6, so the curve crosses the x-axis close to \(x=4\) (and by symmetry at \(x=-4\)).

Step 2: Use the vertex point to test each equation.
At the vertex, \(y=0\). Substitute \(x=4,y=0\) into each option:
Option 1: \(8(4)^2-15(0)^2=8(16)=128\neq169\). Fails.
Option 2: \(9(4)^2-16(0)^2=9(16)=144=144\). Holds exactly.
Option 3: \(|(4-8)(0-15)|=|(-4)(-15)|=60\neq12\). Fails.
Option 4: \(|(4-9)(0-16)|=|(-5)(-16)|=80\neq13\). Fails.

Step 3: Confirm the shape matches a hyperbola.
Equation \(9x^2-16y^2=144\) can be written as \(\frac{x^2}{16}-\frac{y^2}{9}=1\), the standard form of a hyperbola opening along the x-axis with vertices at \(x=\pm4\) (since \(\sqrt{16}=4\)). This matches both the vertex location and the left-right opening shape seen in the picture.

Final Answer:
\[ \boxed{9x^2-16y^2=144} \]