To find the number of tangents that can be drawn from the point \( P(4\sqrt{t}, 3(t-1)) \) to the ellipse given by the equation
\[
\frac{x^2}{16} + \frac{y^2}{9} = 1,
\]
we will follow these steps:
### Step 1: Identify the parameters of the ellipse
The given ellipse has the standard form \(\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1\), where \(a^2 = 16\) and \(b^2 = 9\). Thus, we have:
- \(a = 4\)
- \(b = 3\)
### Step 2: Calculate the distance from the point to the center of the ellipse
The center of the ellipse is at the origin \((0, 0)\). The point \(P\) can be expressed as:
\[
P(4\sqrt{t}, 3(t-1)).
\]
We will calculate the distance \(d\) from the point \(P\) to the center of the ellipse:
\[
d = \sqrt{(4\sqrt{t})^2 + (3(t-1))^2} = \sqrt{16t + 9(t-1)^2}.
\]
### Step 3: Simplify the distance expression
Expanding the distance expression:
\[
d = \sqrt{16t + 9(t^2 - 2t + 1)} = \sqrt{16t + 9t^2 - 18t + 9} = \sqrt{9t^2 - 2t + 9}.
\]
### Step 4: Determine the conditions for the number of tangents
The number of tangents from a point to an ellipse can be determined by comparing the distance \(d\) with the semi-major axis \(a\) and semi-minor axis \(b\):
- If \(d < a\), then there are 0 tangents (point inside the ellipse).
- If \(d = a\), then there is 1 tangent (point on the ellipse).
- If \(d > a\), then there are 2 tangents (point outside the ellipse).
### Step 5: Analyze the expression for \(d\)
We need to find when \(d\) is less than, equal to, or greater than \(a = 4\):
\[
\sqrt{9t^2 - 2t + 9} < 4.
\]
Squaring both sides gives:
\[
9t^2 - 2t + 9 < 16.
\]
This simplifies to:
\[
9t^2 - 2t - 7 < 0.
\]
### Step 6: Solve the quadratic inequality
To solve \(9t^2 - 2t - 7 = 0\), we can use the quadratic formula:
\[
t = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} = \frac{2 \pm \sqrt{(-2)^2 - 4 \cdot 9 \cdot (-7)}}{2 \cdot 9} = \frac{2 \pm \sqrt{4 + 252}}{18} = \frac{2 \pm \sqrt{256}}{18} = \frac{2 \pm 16}{18}.
\]
This gives us the roots:
\[
t_1 = 1, \quad t_2 = -\frac{7}{9}.
\]
### Step 7: Determine the intervals
The quadratic \(9t^2 - 2t - 7\) opens upwards (since the coefficient of \(t^2\) is positive). Thus, it is negative between the roots:
\[
-\frac{7}{9} < t < 1.
\]
### Conclusion
- For \(t < -\frac{7}{9}\) or \(t > 1\), \(d > 4\) (2 tangents).
- For \(t = 1\), \(d = 4\) (1 tangent).
- For \(-\frac{7}{9} < t < 1\), \(d < 4\) (0 tangents).
Thus, the number of tangents that can be drawn from the point \(P(4\sqrt{t}, 3(t-1))\) to the ellipse depends on the value of \(t\):
- **0 tangents** for \(-\frac{7}{9} < t < 1\),
- **1 tangent** for \(t = 1\),
- **2 tangents** for \(t < -\frac{7}{9}\) or \(t > 1\).
