Number of points on the ellipse `x^2/25+y^2/7=1` whose distance from the centre of the ellipse is `2sqrt7` is

Answer

To find the number of points on the ellipse given by the equation \( \frac{x^2}{25} + \frac{y^2}{7} = 1 \) whose distance from the center of the ellipse (0,0) is \( 2\sqrt{7} \), we can follow these steps:

### Step 1: Understand the equation of the ellipse
The equation of the ellipse is given as:
\[ \frac{x^2}{25} + \frac{y^2}{7} = 1 \]
Here, \( a^2 = 25 \) and \( b^2 = 7 \). Therefore, the semi-major axis \( a = 5 \) and the semi-minor axis \( b = \sqrt{7} \).

### Step 2: Calculate the distance from the center
The distance from the center of the ellipse to any point on the ellipse can be described using the distance formula:
\[ d = \sqrt{x^2 + y^2} \]
We need to find points on the ellipse where this distance \( d \) equals \( 2\sqrt{7} \).

### Step 3: Set up the equation for distance
We set the distance equal to \( 2\sqrt{7} \):
\[ \sqrt{x^2 + y^2} = 2\sqrt{7} \]
Squaring both sides gives:
\[ x^2 + y^2 = (2\sqrt{7})^2 = 4 \times 7 = 28 \]

### Step 4: Substitute \( y^2 \) from the ellipse equation
From the ellipse equation, we can express \( y^2 \) in terms of \( x^2 \):
\[ y^2 = 7 \left(1 - \frac{x^2}{25}\right) = 7 - \frac{7x^2}{25} \]
Now, substitute this expression for \( y^2 \) into the distance equation:
\[ x^2 + \left(7 - \frac{7x^2}{25}\right) = 28 \]

### Step 5: Simplify the equation
Combine the terms:
\[ x^2 + 7 - \frac{7x^2}{25} = 28 \]
\[ x^2 - \frac{7x^2}{25} = 28 - 7 \]
\[ x^2 - \frac{7x^2}{25} = 21 \]
To combine the \( x^2 \) terms, find a common denominator:
\[ \frac{25x^2 - 7x^2}{25} = 21 \]
\[ \frac{18x^2}{25} = 21 \]

### Step 6: Solve for \( x^2 \)
Multiply both sides by 25:
\[ 18x^2 = 525 \]
\[ x^2 = \frac{525}{18} = \frac{175}{6} \]

### Step 7: Find \( y^2 \)
Now substitute \( x^2 \) back into the equation for \( y^2 \):
\[ y^2 = 7 - \frac{7 \cdot \frac{175}{6}}{25} \]
Calculating this gives:
\[ y^2 = 7 - \frac{7 \cdot 7}{6} = 7 - \frac{49}{6} = \frac{42}{6} - \frac{49}{6} = -\frac{7}{6} \]

### Step 8: Analyze the result
Since \( y^2 \) cannot be negative, this means there are no real points on the ellipse that are at a distance of \( 2\sqrt{7} \) from the center.

### Conclusion
Thus, the number of points on the ellipse whose distance from the center is \( 2\sqrt{7} \) is:
\[ \text{Number of points} = 0 \]