The mid point of the chord 16x+9y=25 to the ellipse `(x^(2))/(9)+(y^(2))/(16)=1` is

Options

A. (1,-1)
B. (-1,1)
C. (-1,1)
D. (1,1)

Answer

To find the midpoint of the chord \(16x + 9y = 25\) to the ellipse \(\frac{x^2}{9} + \frac{y^2}{16} = 1\), we can follow these steps:

### Step 1: Identify the ellipse and its parameters
The given ellipse is \(\frac{x^2}{9} + \frac{y^2}{16} = 1\). From this equation, we can identify:
- \(a^2 = 9\) (thus \(a = 3\))
- \(b^2 = 16\) (thus \(b = 4\))
- The center of the ellipse is at the origin \((0, 0)\).

**Hint:** Remember that the semi-major axis \(b\) is along the y-axis and the semi-minor axis \(a\) is along the x-axis for this ellipse.

### Step 2: Write the equation of the chord in terms of its midpoint
The equation of the chord can be expressed using the midpoint \((x_1, y_1)\) as follows:
\[ \frac{x x_1}{9} + \frac{y y_1}{16} = \frac{x_1^2}{9} + \frac{y_1^2}{16} \]
This is derived from the property of chords in conics.

**Hint:** The equation of the chord through a point \((x_1, y_1)\) on the ellipse can be derived from the tangent line at that point.

### Step 3: Substitute the equation of the chord
We have the chord equation \(16x + 9y = 25\). We can rewrite this in the form:
\[ \frac{x x_1}{9} + \frac{y y_1}{16} = \frac{x_1^2}{9} + \frac{y_1^2}{16} \]
Now we need to equate this with the chord equation.

**Hint:** Rearranging the equation of the chord can help in identifying the relationship between \(x_1\) and \(y_1\).

### Step 4: Solve for the midpoint coordinates
To find the coordinates of the midpoint, we need to express \(x_1\) and \(y_1\) in terms of the chord equation.

From the chord equation:
\[ 16x + 9y = 25 \]
If we set \(x = 0\), we find \(y = \frac{25}{9}\), and if we set \(y = 0\), we find \(x = \frac{25}{16}\). Thus, the points where the chord intersects the axes are \((0, \frac{25}{9})\) and \((\frac{25}{16}, 0)\).

**Hint:** Finding intercepts can help visualize the chord and its midpoint.

### Step 5: Find the midpoint of the chord
The midpoint \(M\) of the chord can be calculated as follows:
\[ M = \left(\frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2}\right) \]
Substituting the intercepts:
\[ M = \left(\frac{0 + \frac{25}{16}}{2}, \frac{\frac{25}{9} + 0}{2}\right) = \left(\frac{25}{32}, \frac{25}{18}\right) \]

**Hint:** The midpoint formula is essential for finding the average of the coordinates of the endpoints.

### Final Answer
Thus, the midpoint of the chord \(16x + 9y = 25\) to the ellipse \(\frac{x^2}{9} + \frac{y^2}{16} = 1\) is:
\[ \left(\frac{25}{32}, \frac{25}{18}\right) \]