Statement 1 The sum of the focal distances of a point on the ellipse `4x^(2)+5y^(2)-16x-30y41=0 is 2sqrt5`.
Statement 2 The equation `4x^(2)+5y^(2)-16x-30y+41=0` can be expressed as `4(x-2)^(2)+5(y-3)^(2)=20`.

Options

A. Statement I is true, statement II is true: statement II is a correct explanation for statement I
B. Statement I is true, statement II is true, statement II is not a correct explanation for statement I
C. statement I is true, statement II is false
D. statement I is false, statement II is true

Answer

To solve the problem step by step, we will analyze the given statements and derive the necessary conclusions.

### Step 1: Rewrite the given equation
The given equation is:
\[ 4x^2 + 5y^2 - 16x - 30y + 41 = 0 \]

We can rearrange it to isolate the constant on one side:
\[ 4x^2 + 5y^2 - 16x - 30y = -41 \]