If the vertices of the parabola be at `(-2,0)` and `(2,0)` and one of the foci be at `(-3,0)` then which one of the following points does not lie on the hyperbola? (a) `(-6, 2sqrt(10))` (b) `(2sqrt6,5)` (c) `(4, sqrt(15))` (d) `(6, 5sqrt2)`

Options

A. `( 4, sqrt(15))`
B. `(-6,2sqrt(10))`
C. `(6,5sqrt(2))`
D. `(2, sqrt(6),5)`

Answer

To solve the problem, we need to find the equation of the hyperbola given its vertices and one of its foci. The vertices are at (-2, 0) and (2, 0), and one of the foci is at (-3, 0).

### Step 1: Identify the parameters of the hyperbola
The vertices of the hyperbola are given as (-2, 0) and (2, 0). This indicates that the hyperbola is centered at the origin (0, 0) and opens horizontally. The distance from the center to each vertex is denoted as 'a'.

From the vertices:
- The distance \( a = 2 \)

### Step 2: Find the value of 'c'
The foci of the hyperbola are given as (-3, 0). The distance from the center to the foci is denoted as 'c'.

From the foci:
- The distance \( c = 3 \)

### Step 3: Use the relationship between a, b, and c
For hyperbolas, the relationship between 'a', 'b', and 'c' is given by:
\[ c^2 = a^2 + b^2 \]

We already have:
- \( a = 2 \) so \( a^2 = 4 \)
- \( c = 3 \) so \( c^2 = 9 \)

Substituting these values into the equation:
\[ 9 = 4 + b^2 \]
\[ b^2 = 9 - 4 = 5 \]

### Step 4: Write the equation of the hyperbola
Now that we have \( a^2 \) and \( b^2 \), we can write the standard form of the hyperbola:
\[ \frac{x^2}{a^2} - \frac{y^2}{b^2} = 1 \]
Substituting the values:
\[ \frac{x^2}{4} - \frac{y^2}{5} = 1 \]

### Step 5: Check each point to see if it lies on the hyperbola
We will now check each of the given points to see if they satisfy the hyperbola equation.

1. **Point (-6, 2√10)**:
\[ \frac{(-6)^2}{4} - \frac{(2\sqrt{10})^2}{5} = \frac{36}{4} - \frac{40}{5} = 9 - 8 = 1 \quad \text{(lies on the hyperbola)} \]

2. **Point (2√6, 5)**:
\[ \frac{(2\sqrt{6})^2}{4} - \frac{(5)^2}{5} = \frac{24}{4} - \frac{25}{5} = 6 - 5 = 1 \quad \text{(lies on the hyperbola)} \]

3. **Point (4, √15)**:
\[ \frac{(4)^2}{4} - \frac{(\sqrt{15})^2}{5} = \frac{16}{4} - \frac{15}{5} = 4 - 3 = 1 \quad \text{(lies on the hyperbola)} \]

4. **Point (6, 5√2)**:
\[ \frac{(6)^2}{4} - \frac{(5\sqrt{2})^2}{5} = \frac{36}{4} - \frac{50}{5} = 9 - 10 = -1 \quad \text{(does not lie on the hyperbola)} \]

### Conclusion
The point that does not lie on the hyperbola is **(6, 5√2)**.