The point of contact `4x-5y+25=0` with the ellipse `9x^(2)+25y^(2)=225` is

Options

A. `(-4,9/5)`
B. `(-4,3/5)`
C. `(4,-3)`
D. `(-5,2)`

Answer

To find the point of contact of the line \(4x - 5y + 25 = 0\) with the ellipse \(9x^2 + 25y^2 = 225\), we can follow these steps:

### Step 1: Rewrite the equation of the ellipse in standard form
The given ellipse equation is:
\[ 9x^2 + 25y^2 = 225 \]
Dividing the entire equation by 225, we get:
\[ \frac{x^2}{25} + \frac{y^2}{9} = 1 \]
This shows that \(a^2 = 25\) and \(b^2 = 9\), where \(a = 5\) and \(b = 3\).

**Hint:** Always convert the equation of the ellipse to the standard form \(\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1\).

### Step 2: Identify the point of contact
Let the point of contact be \((h, k)\). The equation of the tangent to the ellipse at the point \((h, k)\) is given by:
\[ \frac{hx}{25} + \frac{ky}{9} = 1 \]

**Hint:** Remember that the tangent line at a point on the ellipse can be expressed in terms of the coordinates of that point.

### Step 3: Set the tangent equation equal to the given line
We know the tangent equation must match the given line equation:
\[ \frac{hx}{25} + \frac{ky}{9} = 1 \]
We can rewrite this as:
\[ hx + \frac{25ky}{9} = 25 \]
Now, we compare this with the line equation \(4x - 5y + 25 = 0\) or equivalently \(4x + 5y = 25\).

**Hint:** To find the coefficients that match, rearranging the equation can help in comparison.

### Step 4: Compare coefficients
From the equations:
\[ hx + \frac{25ky}{9} = 25 \quad \text{and} \quad 4x + 5y = 25 \]
We can equate the coefficients of \(x\) and \(y\):
1. \(h = 4\)
2. \(\frac{25k}{9} = 5\)

**Hint:** Setting the coefficients equal allows us to solve for \(h\) and \(k\).

### Step 5: Solve for \(k\)
From \(\frac{25k}{9} = 5\):
\[ 25k = 45 \implies k = \frac{45}{25} = \frac{9}{5} \]

**Hint:** Simplifying fractions can help in finding the values easily.

### Step 6: Conclusion
Thus, the point of contact \((h, k)\) is:
\[ (h, k) = (4, \frac{9}{5}) \]

### Final Answer
The point of contact of the line \(4x - 5y + 25 = 0\) with the ellipse \(9x^2 + 25y^2 = 225\) is:
\[ \boxed{(4, \frac{9}{5})} \]