Latus rectum of the parabola whose focus is (3, 4) and whose tangent at vertex has the equation x + y = 7 + 5√2 is:

• A) 5
• B) 10
• C) 20
• D) 15

To find the latus rectum of the parabola with a given focus and tangent, we first need to determine the equation of the parabola. The focus is at the point (3, 4), and the tangent at the vertex is given by the equation x + y = 7 + 5√2.

Step 1: Identify the Vertex

The vertex of the parabola lies on the tangent line. To find the vertex, we can rearrange the tangent equation:

x + y - 5√2 = 7

From this, we can express y in terms of x:

y = -x + 7 + 5√2

To find the vertex, we need to determine where this line intersects the axis of symmetry of the parabola.

Step 2: Determine the Orientation

Since the focus is above the vertex (y-coordinate of the focus is greater than the y-coordinate of the vertex), the parabola opens upwards. The standard form of a vertical parabola is:

(x - h)² = 4p(y - k)

where (h, k) is the vertex and p is the distance from the vertex to the focus.

Step 3: Calculate p

The distance p can be calculated as the distance from the vertex to the focus. If we denote the vertex as (h, k), then:

• Focus: (3, 4)
• Vertex: (h, k)
• p = |k - 4|

Step 4: Latus Rectum Calculation

The length of the latus rectum of a parabola is given by the formula:

Length of Latus Rectum = 4p

To find the correct value, we need to determine p based on the vertex we find from the tangent line. After calculating, we find that:

p = 5

Thus, the length of the latus rectum is:

4p = 4 * 5 = 20

Final Answer

The latus rectum of the parabola is 20, so the correct option is C) 20.