The vertices of the triangle ABC are `A(0, 0), B(3, 0) and C(3, 4)`, where A and C are foci of an ellipse and B lies on the ellipse. If the length of the latus rectum of the ellipse is `(12)/(p)` units, then the vlaue of p is

Answer

To solve the problem, we need to find the value of \( p \) given the vertices of triangle ABC and the properties of the ellipse formed by points A and C as foci, and point B lying on the ellipse.

### Step-by-Step Solution:

1. **Identify the Coordinates of Points:**
- The coordinates of the points are:
- \( A(0, 0) \)
- \( B(3, 0) \)
- \( C(3, 4) \)

2. **Determine the Foci and Major Axis:**
- The foci of the ellipse are at points A and C, which are \( (0, 0) \) and \( (3, 4) \) respectively.
- The distance between the foci \( A \) and \( C \) can be calculated using the distance formula:
\[ AC = \sqrt{(3 - 0)^2 + (4 - 0)^2} = \sqrt{9 + 16} = \sqrt{25} = 5 \]
- The distance between the foci is \( 2c \), so \( c = \frac{5}{2} \).

3. **Use the Property of the Ellipse:**
- The point \( B \) lies on the ellipse, and the sum of the distances from any point on the ellipse to the two foci is equal to the length of the major axis \( 2a \).
- Calculate the distances \( BA \) and \( BC \):
- Distance \( BA \) from \( B(3, 0) \) to \( A(0, 0) \):
\[ BA = \sqrt{(3 - 0)^2 + (0 - 0)^2} = 3 \]
- Distance \( BC \) from \( B(3, 0) \) to \( C(3, 4) \):
\[ BC = \sqrt{(3 - 3)^2 + (0 - 4)^2} = 4 \]
- Thus, the sum of the distances is:
\[ BA + BC = 3 + 4 = 7 \]
- Therefore, we have:
\[ 2a = 7 \implies a = \frac{7}{2} \]

4. **Find the Length of the Latus Rectum:**
- The length of the latus rectum \( L \) of the ellipse is given by the formula:
\[ L = \frac{2b^2}{a} \]
- To find \( b^2 \), we use the relationship \( c^2 = a^2 - b^2 \):
- We know \( c = \frac{5}{2} \) and \( a = \frac{7}{2} \):
\[ c^2 = \left(\frac{5}{2}\right)^2 = \frac{25}{4} \]
\[ a^2 = \left(\frac{7}{2}\right)^2 = \frac{49}{4} \]
- Therefore:
\[ b^2 = a^2 - c^2 = \frac{49}{4} - \frac{25}{4} = \frac{24}{4} = 6 \]

5. **Substitute Values into the Latus Rectum Formula:**
- Now substituting \( b^2 \) and \( a \) into the latus rectum formula:
\[ L = \frac{2b^2}{a} = \frac{2 \times 6}{\frac{7}{2}} = \frac{12}{\frac{7}{2}} = \frac{12 \times 2}{7} = \frac{24}{7} \]

6. **Relate to Given Length of Latus Rectum:**
- We are given that the length of the latus rectum is \( \frac{12}{p} \):
\[ \frac{24}{7} = \frac{12}{p} \]
- Cross-multiplying gives:
\[ 24p = 84 \implies p = \frac{84}{24} = \frac{7}{2} \]

### Final Answer:
Thus, the value of \( p \) is \( \frac{7}{2} \).

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