Question

The equations of the tangent to the curve \(x^2+y^2 = 10\), where the tangent is parallel to the line \(2x+y-1 = 0\), are

• A. \(2x+y = \pm \sqrt{2}\)
• B. \(2x+y = \pm 5\sqrt{2}\)
• C. \(2x+y = \pm 2\sqrt{2}\)
• D. \(2x+y = \pm 3\sqrt{2}\)

Hint:

A tangent y = mx + c to a circle of radius r needs c^2 = r^2(1 + m^2).

Answer 1

The Correct Option is B

Step 1: Distance idea:
A line is tangent when its distance from the centre equals the radius $\sqrt{10}$.

Step 2: Parallel family:
Lines parallel to $2x+y-1=0$ are $2x + y = k$, at distance $|k|/\sqrt5$ from the origin.

Step 3: Solve:
$|k|/\sqrt5 = \sqrt{10} \Rightarrow |k| = \sqrt{50} = 5\sqrt2$. So $2x+y = \pm5\sqrt2$, option (B).

Final Answer:
Tangents are 2x + y = plus or minus 5 root 2. \[ \boxed{\text{(B) }2x+y=\pm5\sqrt2} \]