If the ellipse `9x^(2)+16y^(2)=144` intercepts the line `3x+ 4y = 12`. then what is the length of the chord so formed?

Options

A. 5 units
B. 6 units
C. 8 units
D. 10 units

Answer

To find the length of the chord formed by the intersection of the ellipse \(9x^2 + 16y^2 = 144\) and the line \(3x + 4y = 12\), we can follow these steps:

### Step 1: Rewrite the equations
First, we rewrite the equations of the ellipse and the line in a more manageable form.

The equation of the ellipse can be rewritten as:
\[ \frac{x^2}{16} + \frac{y^2}{9} = 1 \]
This shows that the semi-major axis \(a = 4\) and the semi-minor axis \(b = 3\).

The equation of the line can be rearranged to express \(y\) in terms of \(x\):
\[ 4y = 12 - 3x \implies y = 3 - \frac{3}{4}x \]

### Step 2: Substitute the line equation into the ellipse equation
Next, we substitute the expression for \(y\) from the line equation into the ellipse equation:
\[ 9x^2 + 16\left(3 - \frac{3}{4}x\right)^2 = 144 \]
Expanding the equation:
\[ 9x^2 + 16\left(9 - \frac{18}{4}x + \frac{9}{16}x^2\right) = 144 \]
\[ 9x^2 + 144 - 72x + 9x^2 = 144 \]
Combining like terms:
\[ 18x^2 - 72x + 144 - 144 = 0 \]
This simplifies to:
\[ 18x^2 - 72x = 0 \]
Factoring out \(18x\):
\[ 18x(x - 4) = 0 \]

### Step 3: Solve for \(x\)
Setting each factor to zero gives us:
\[ x = 0 \quad \text{or} \quad x = 4 \]

### Step 4: Find corresponding \(y\) values
Now we can find the corresponding \(y\) values using the line equation:
1. For \(x = 0\):
\[ y = 3 - \frac{3}{4}(0) = 3 \implies (0, 3) \]
2. For \(x = 4\):
\[ y = 3 - \frac{3}{4}(4) = 3 - 3 = 0 \implies (4, 0) \]

### Step 5: Calculate the length of the chord
Now we have the points of intersection: \((0, 3)\) and \((4, 0)\). We can use the distance formula to find the length of the chord:
\[ \text{Length} = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \]
Substituting the values:
\[ \text{Length} = \sqrt{(4 - 0)^2 + (0 - 3)^2} = \sqrt{4^2 + (-3)^2} = \sqrt{16 + 9} = \sqrt{25} = 5 \]

### Final Answer
The length of the chord formed is \(5\).