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Step-by-Step Solution
Step 1: Identify the key details of the hyperbola
• The vertices of the hyperbola are given as $(-2,0)$ and $(2,0)$. • One of the foci is given as $(-3,0)$. • From the vertices, we can see that the center of the hyperbola is at the origin $(0,0)$ and its transverse axis is along the x-axis.
Step 2: Recall the standard form of the hyperbola
For a hyperbola centered at the origin with transverse axis along the x-axis, the standard form is Here, • $a$ is the semi-major axis (distance from center to vertex), • $c$ is the semi-focal distance (distance from center to focus), • $b$ is related by the equation $c^2 = a^2 + b^2$.
Step 3: Find the values of $a$ and $b$
• Because the vertices are at $(-2,0)$ and $(2,0)$, it follows that $a = 2$. • One of the foci is at $(-3,0)$, so $c = 3$. • Use $c^2 = a^2 + b^2$ to find $b^2$: • Thus, $b = \sqrt{5}$.
Step 4: Write the equation of the hyperbola
Substituting $a = 2$ and $b = \sqrt{5}$ into the standard form:
Step 5: Check which option does not satisfy the hyperbola equation
We are told that the point $\left(6,\,5\sqrt{2}\right)$ is the correct answer (i.e., does not lie on the hyperbola). Verify by substituting $x=6$ and $y=5\sqrt{2}$ into the equation: Since the resulting value is $-1$ instead of $1$, $\left(6,\,5\sqrt{2}\right)$ does not lie on the hyperbola.
Final Answer
The point $\bigl(6,\,5\sqrt{2}\bigr)$ does not lie on the given hyperbola.
Step-by-Step Solution
Step 1: Identify the key details of the hyperbola
• The vertices of the hyperbola are given as $(-2,0)$ and $(2,0)$. • One of the foci is given as $(-3,0)$. • From the vertices, we can see that the center of the hyperbola is at the origin $(0,0)$ and its transverse axis is along the x-axis.
Step 2: Recall the standard form of the hyperbola
For a hyperbola centered at the origin with transverse axis along the x-axis, the standard form is x2a2-y2b2=1. \frac{x^2}{a^2} - \frac{y^2}{b^2} = 1. Here, • $a$ is the semi-major axis (distance from center to vertex), • $c$ is the semi-focal distance (distance from center to focus), • $b$ is related by the equation $c^2 = a^2 + b^2$.
Step 3: Find the values of $a$ and $b$
• Because the vertices are at $(-2,0)$ and $(2,0)$, it follows that $a = 2$. • One of the foci is at $(-3,0)$, so $c = 3$. • Use $c^2 = a^2 + b^2$ to find $b^2$: 32=22+b2 ⟹ 9=4+b2 ⟹ b2=5. 3^2 = 2^2 + b^2 \quad \Longrightarrow \quad 9 = 4 + b^2 \quad \Longrightarrow \quad b^2 = 5. • Thus, $b = \sqrt{5}$.
Step 4: Write the equation of the hyperbola
Substituting $a = 2$ and $b = \sqrt{5}$ into the standard form: x24-y25=1. \frac{x^2}{4} - \frac{y^2}{5} = 1.
Step 5: Check which option does not satisfy the hyperbola equation
We are told that the point $\left(6,\,5\sqrt{2}\right)$ is the correct answer (i.e., does not lie on the hyperbola). Verify by substituting $x=6$ and $y=5\sqrt{2}$ into the equation: 624-5225=364-25·25=9-10=-1 ≠ 1. \frac{6^2}{4} - \frac{\left(5\sqrt{2}\right)^2}{5} = \frac{36}{4} - \frac{25 \cdot 2}{5} = 9 - 10 = -1 \,\neq\, 1. Since the resulting value is $-1$ instead of $1$, $\left(6,\,5\sqrt{2}\right)$ does not lie on the hyperbola.
Final Answer
The point $\bigl(6,\,5\sqrt{2}\bigr)$ does not lie on the given hyperbola.
