If y=mx+5 is a tangent to the ellipse `4x^2+25y^2=100`, then find the value of `100m^2`.

Answer

To solve the problem step-by-step, we need to find the value of \(100m^2\) given that the line \(y = mx + 5\) is a tangent to the ellipse \(4x^2 + 25y^2 = 100\).

### Step 1: Rewrite the equation of the ellipse
We start with the equation of the ellipse:
\[ 4x^2 + 25y^2 = 100 \]
Divide the entire equation by 100 to put it in standard form:
\[ \frac{4x^2}{100} + \frac{25y^2}{100} = 1 \]
This simplifies to:
\[ \frac{x^2}{25} + \frac{y^2}{4} = 1 \]
From this, we can identify \(a^2 = 25\) and \(b^2 = 4\).

### Step 2: Identify the values of \(a\) and \(b\)
Taking the square roots, we find:
\[ a = 5 \quad \text{and} \quad b = 2 \]

### Step 3: Use the tangent condition
For a line \(y = mx + c\) to be tangent to the ellipse, the condition is:
\[ c = \pm \sqrt{a^2 m^2 + b^2} \]
In our case, \(c = 5\), so we set up the equation:
\[ 5 = \pm \sqrt{25m^2 + 4} \]

### Step 4: Square both sides
We will consider the positive case first:
\[ 5 = \sqrt{25m^2 + 4} \]
Squaring both sides gives:
\[ 25 = 25m^2 + 4 \]

### Step 5: Solve for \(m^2\)
Rearranging the equation:
\[ 25m^2 = 25 - 4 \]
\[ 25m^2 = 21 \]
\[ m^2 = \frac{21}{25} \]

### Step 6: Find \(100m^2\)
Now, we multiply \(m^2\) by 100:
\[ 100m^2 = 100 \times \frac{21}{25} \]
This simplifies to:
\[ 100m^2 = 4 \times 21 = 84 \]

### Final Answer
Thus, the value of \(100m^2\) is:
\[ \boxed{84} \]