1. Key Concepts and Formulas

This problem requires us to determine the equation of a hyperbola by leveraging its properties and then verify which of the given points does not lie on it. We will utilize the standard forms and fundamental relationships for both ellipses and hyperbolas, particularly focusing on their foci and axis lengths.

• Standard Equation of an Ellipse: For an ellipse centered at the origin with its major axis along the x-axis (i.e., $a_e > b_e$), the equation is given by: $\frac{x^2}{a_e^2} + \frac{y^2}{b_e^2} = 1$ The foci of such an ellipse are located at $(\pm c_e, 0)$, where $c_e$ is related to $a_e$ and $b_e$ by the equation $c_e^2 = a_e^2 - b_e^2$.

• Standard Equation of a Hyperbola: For a hyperbola centered at the origin with its transverse axis along the x-axis, the equation is given by: $\frac{x^2}{a_h^2} - \frac{y^2}{b_h^2} = 1$ The foci of such a hyperbola are located at $(\pm c_h, 0)$, where $c_h$ is related to $a_h$ and $b_h$ by the equation $c_h^2 = a_h^2 + b_h^2$. The length of the transverse axis is $2a_h$.

2. Step-by-Step Working

Step 1: Analyze the given Ellipse to find its Foci.

The equation of the given ellipse is $3x^2 + 4y^2 = 12$. To identify its properties, we first convert it into the standard form $\frac{x^2}{a_e^2} + \frac{y^2}{b_e^2} = 1$.

• Why this step? The standard form allows us to directly identify the values of $a_e^2$ and $b_e^2$, which are crucial for finding the foci.

Divide the entire equation by 12: $\frac{3x^2}{12} + \frac{4y^2}{12} = \frac{12}{12}$ $\frac{x^2}{4} + \frac{y^2}{3} = 1$

Comparing this with the standard form $\frac{x^2}{a_e^2} + \frac{y^2}{b_e^2} = 1$, we have: $a_e^2 = 4 \implies a_e = 2$ $b_e^2 = 3 \implies b_e = \sqrt{3}$

Since $a_e^2 > b_e^2$ (i.e., $4 > 3$), the major axis of the ellipse lies along the x-axis. Now, we find the distance of the foci from the center, $c_e$. Using the relation $c_e^2 = a_e^2 - b_e^2$: $c_e^2 = 4 - 3$ $c_e^2 = 1$ $c_e = 1$

The foci of the ellipse are $(\pm c_e, 0)$, which are $(\pm 1, 0)$.

Step 2: Determine the Foci and Transverse Axis Orientation of the Hyperbola.

The problem states that the hyperbola has the same foci as the ellipse. Therefore, the foci of the hyperbola are also $(\pm 1, 0)$.

• Why this step? The foci's location tells us the center and the orientation of the transverse axis of the hyperbola. Since the foci are on the x-axis, the hyperbola is centered at the origin and its transverse axis is also along the x-axis. This allows us to use the standard form $\frac{x^2}{a_h^2} - \frac{y^2}{b_h^2} = 1$.

From the foci, we know that for the hyperbola, $c_h = 1$. So, $c_h^2 = 1$.

Step 3: Use the Length of the Transverse Axis to find $a_h$.

The length of the transverse axis of the hyperbola is given as $\sqrt{2}$. For a hyperbola with its transverse axis along the x-axis, the length of the transverse axis is $2a_h$.

• Why this step? The length of the transverse axis directly gives us the value of $a_h$, which is one of the parameters needed for the hyperbola's equation.

So, $2a_h = \sqrt{2}$. $a_h = \frac{\sqrt{2}}{2} = \frac{1}{\sqrt{2}}$ $a_h^2 = \left( \frac{1}{\sqrt{2}} \right)^2 = \frac{1}{2}$

Step 4: Calculate $b_h^2$ for the Hyperbola.

For a hyperbola with its transverse axis along the x-axis, the relationship between $a_h, b_h,$ and $c_h$ is $c_h^2 = a_h^2 + b_h^2$.

• Why this step? We have $c_h^2$ (from the foci) and $a_h^2$ (from the transverse axis length). This relation allows us to find $b_h^2$, the final parameter needed to write the hyperbola's equation.

Substitute the values of $c_h^2 = 1$ and $a_h^2 = \frac{1}{2}$: $1 = \frac{1}{2} + b_h^2$ $b_h^2 = 1 - \frac{1}{2}$ $b_h^2 = \frac{1}{2}$

Step 5: Write the Equation of the Hyperbola.

Now that we have $a_h^2 = \frac{1}{2}$ and $b_h^2 = \frac{1}{2}$, we can write the equation of the hyperbola using the standard form $\frac{x^2}{a_h^2} - \frac{y^2}{b_h^2} = 1$: $\frac{x^2}{\frac{1}{2}} - \frac{y^2}{\frac{1}{2}} = 1$ $2x^2 - 2y^2 = 1$

Step 6: Test the Given Points to find which one does not lie on the Hyperbola.

We will substitute the coordinates of each option into the hyperbola's equation $2x^2 - 2y^2 = 1$ and check if the equation holds true.

• Why this step? A point lies on a curve if and only if its coordinates satisfy the equation of the curve. We are looking for the point that does not satisfy the equation.

(A) Point: $\left( {1, - {1 \over {\sqrt 2 }}} \right)$ Substitute $x=1$ and $y = - \frac{1}{\sqrt{2}}$: $2(1)^2 - 2\left( - \frac{1}{\sqrt{2}} \right)^2 = 2(1) - 2\left( \frac{1}{2} \right)$ $= 2 - 1 = 1$ Since $1 = 1$, this point lies on the hyperbola.

(B) Point: $\left( {\sqrt {{3 \over 2}} ,{1 \over {\sqrt 2 }}} \right)$ Substitute $x=\sqrt{\frac{3}{2}}$ and $y = \frac{1}{\sqrt{2}}$: $2\left( \sqrt{\frac{3}{2}} \right)^2 - 2\left( \frac{1}{\sqrt{2}} \right)^2 = 2\left( \frac{3}{2} \right) - 2\left( \frac{1}{2} \right)$ $= 3 - 1 = 2$ Since $2 \ne 1$, this point does not lie on the hyperbola.

(C) Point: $\left( { - \sqrt {{3 \over 2}} ,1} \right)$ Substitute $x= - \sqrt{\frac{3}{2}}$ and $y = 1$: $2\left( - \sqrt{\frac{3}{2}} \right)^2 - 2(1)^2 = 2\left( \frac{3}{2} \right) - 2(1)$ $= 3 - 2 = 1$ Since $1 = 1$, this point lies on the hyperbola.

(D) Point: $\left( {{1 \over {\sqrt 2 }},0} \right)$ Substitute $x=\frac{1}{\sqrt{2}}$ and $y = 0$: $2\left( \frac{1}{\sqrt{2}} \right)^2 - 2(0)^2 = 2\left( \frac{1}{2} \right) - 0$ $= 1 - 0 = 1$ Since $1 = 1$, this point lies on the hyperbola.

From the above checks, point (B) does not satisfy the equation of the hyperbola.

3. Relevant Tips and Common Mistakes

• Distinguish $a^2$ and $b^2$ for Ellipse vs. Hyperbola:
  • For an ellipse, $a_e^2$ is always the larger denominator (under the major axis variable).
  • For a hyperbola, $a_h^2$ is always the denominator under the positive term (which defines the transverse axis). The values of $a_h$ and $b_h$ are not necessarily related by size.
• Foci Relation: Remember the correct relation for $c^2$.
  • Ellipse: $c_e^2 = a_e^2 - b_e^2$ (subtraction, as $c_e < a_e$)
  • Hyperbola: $c_h^2 = a_h^2 + b_h^2$ (addition, as $c_h > a_h$)
• Transverse Axis: The length of the transverse axis is $2a_h$, not $2b_h$. Make sure to correctly identify $a_h$ from the given length.
• Careful with Arithmetic: Double-check calculations, especially when dealing with square roots and fractions.

4. Summary and Key Takeaway

We successfully determined the equation of the hyperbola by first finding the common foci from the given ellipse. The orientation of the transverse axis was inferred from the foci's location. Using the given transverse axis length, we found $a_h^2$. Finally, we used the fundamental relation $c_h^2 = a_h^2 + b_h^2$ to find $b_h^2$ and complete the hyperbola's equation $2x^2 - 2y^2 = 1$. By substituting each given point into this equation, we identified that point (B) $\left( {\sqrt {{3 \over 2}} ,{1 \over {\sqrt 2 }}} \right)$ does not satisfy the equation, hence it does not lie on the hyperbola.

The final answer is $\boxed{\text{(B)}}$.