Angle between the tangents drawn from the point (5,4) to the ellipse `(x^(2))/(25)+(y^(2))/(16) = 1` is

Options

A. `45^(@)`
B. `60^(@)`
C. `90^(@)`
D. `120^(@)`

Answer

To find the angle between the tangents drawn from the point (5, 4) to the ellipse given by the equation \(\frac{x^2}{25} + \frac{y^2}{16} = 1\), we can follow these steps:

### Step 1: Identify the parameters of the ellipse
The equation of the ellipse is in the standard form \(\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1\), where:
- \(a^2 = 25\) (thus, \(a = 5\))
- \(b^2 = 16\) (thus, \(b = 4\))

### Step 2: Use the formula for the angle between the tangents
The angle \(\theta\) between the tangents drawn from a point \((x_1, y_1)\) to the ellipse can be calculated using the formula:
\[ \tan(\theta) = \frac{2\sqrt{h}}{a^2 + b^2 - (x_1^2 + y_1^2)} \]
where \(h = \frac{x_1^2}{a^2} + \frac{y_1^2}{b^2} - 1\).

### Step 3: Calculate \(h\)
Substituting \(x_1 = 5\) and \(y_1 = 4\):
\[ h = \frac{5^2}{25} + \frac{4^2}{16} - 1 = \frac{25}{25} + \frac{16}{16} - 1 = 1 + 1 - 1 = 1 \]

### Step 4: Calculate \(a^2 + b^2 - (x_1^2 + y_1^2)\)
Now, we calculate:
\[ x_1^2 + y_1^2 = 5^2 + 4^2 = 25 + 16 = 41 \]
Thus,
\[ a^2 + b^2 = 25 + 16 = 41 \]
So,
\[ a^2 + b^2 - (x_1^2 + y_1^2) = 41 - 41 = 0 \]

### Step 5: Substitute into the tangent formula
Now substituting into the tangent formula:
\[ \tan(\theta) = \frac{2\sqrt{1}}{0} = \text{undefined} \]

### Step 6: Determine the angle
Since \(\tan(\theta)\) is undefined, this implies that \(\theta = 90^\circ\).

### Conclusion
The angle between the tangents drawn from the point (5, 4) to the ellipse is \(90^\circ\).

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