The number of tangents (real) that can be drawn to the ellipse `5x^(2)+7y^(2)=40` passing through (3,5) is

Options

A. 4
B. 3
C. 1
D. 2

Answer

To find the number of tangents that can be drawn to the ellipse \(5x^2 + 7y^2 = 40\) from the point (3, 5), we will follow these steps:

### Step 1: Rewrite the equation of the ellipse
The equation of the ellipse is given as:
\[ 5x^2 + 7y^2 = 40 \]
We can rewrite this in standard form:
\[ \frac{x^2}{8} + \frac{y^2}{\frac{40}{7}} = 1 \]
This shows that the ellipse is centered at the origin with semi-major and semi-minor axes.

### Step 2: Determine the condition for the point (3, 5)
To determine how many tangents can be drawn from the point (3, 5) to the ellipse, we use the concept of the discriminant of the quadratic equation formed by substituting the point into the general equation of the ellipse.

### Step 3: Substitute the point into the ellipse equation
We substitute \(x = 3\) and \(y = 5\) into the equation of the ellipse:
\[ S_1 = 5(3^2) + 7(5^2) - 40 \]
Calculating this:
\[ S_1 = 5(9) + 7(25) - 40 = 45 + 175 - 40 = 180 \]

### Step 4: Analyze the value of \(S_1\)
Now we check the value of \(S_1\):
\[ S_1 = 180 \]
Since \(S_1 > 0\), this indicates that the point (3, 5) lies outside the ellipse.

### Step 5: Conclusion about the number of tangents
If the point lies outside the ellipse, then there are two tangents that can be drawn from that point to the ellipse.

Thus, the number of tangents that can be drawn to the ellipse \(5x^2 + 7y^2 = 40\) from the point (3, 5) is:
\[ \text{Number of tangents} = 2 \]

### Final Answer:
The number of tangents (real) that can be drawn to the ellipse from the point (3, 5) is **2**.

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