Quadilateral KLMN is enlarged by a factor of p.Find the largest value of p for which the image of KLMN is contained in the circle centered at (0;0) with radius 1

To find the largest value of p for which the image of quadrilateral KLMN is contained in the circle centered at (0,0) with radius 1, we need to consider the vertices of the quadrilateral and how they change after the enlargement.

Let's assume that the coordinates of the vertices of quadrilateral KLMN are (x₁, y₁), (x₂, y₂), (x₃, y₃), and (x₄, y₄).

When the quadrilateral is enlarged by a factor of p, the coordinates of the vertices are multiplied by p. So, the new coordinates of the vertices become (px₁, py₁), (px₂, py₂), (px₃, py₃), and (px₄, py₄).

For the image of the quadrilateral to be contained within the circle centered at (0,0) with radius 1, we need to ensure that the maximum distance from any vertex of the quadrilateral to the origin (0,0) is less than or equal to the radius of the circle, which is 1.

So, we need to find the maximum distance from any vertex of the enlarged quadrilateral to the origin, and the maximum distance is given by the formula:

d = sqrt(x² + y²)

For the maximum distance to be less than or equal to 1, we have:

sqrt((px₁)² + (py₁)²) ≤ 1
sqrt((px₂)² + (py₂)²) ≤ 1
sqrt((px₃)² + (py₃)²) ≤ 1
sqrt((px₄)² + (py₄)²) ≤ 1

Now, let's simplify these inequalities:

(p²x₁² + p²y₁²) ≤ 1
(p²x₂² + p²y₂²) ≤ 1
(p²x₃² + p²y₃²) ≤ 1
(p²x₄² + p²y₄²) ≤ 1

We can divide both sides of the inequalities by p²:

x₁² + y₁²/p² ≤ 1/p²
x₂² + y₂²/p² ≤ 1/p²
x₃² + y₃²/p² ≤ 1/p²
x₄² + y₄²/p² ≤ 1/p²

Since p is a positive value, we can multiply both sides of the inequalities by p²:

x₁²p² + y₁² ≤ 1
x₂²p² + y₂² ≤ 1
x₃²p² + y₃² ≤ 1
x₄²p² + y₄² ≤ 1

To find the largest value of p, we need to maximize p². Since all the terms on the left side of the inequalities are positive, the largest value of p² will occur when the left side of each inequality is equal to 1.

Therefore, the largest value of p is given by:

p = 1 / sqrt(max(x₁² + y₁², x₂² + y₂², x₃² + y₃², x₄² + y₄²))