Question

If the lines joining the origin to the points of intersection of a line $L$ and $x^2 + y^2 = 4$ are the coordinate axes, then the equation of the line $L$ is

• A. $x + y = 2$
• B. $x + y = 4$
• C. $x + y = 1$
• D. $x + y = 0$

Hint:

Homogenization Insight: If two points lie on a circle and their joining lines with origin are coordinate axes, the intersecting line must pass through intercepts $(\pm r, 0)$ and $(0, \pm r)$ of the circle.

Answer 1

The Correct Option is A

Let $L$ intersect the circle $x^2 + y^2 = 4$ at points lying on $x$-axis and $y$-axis. Points: $(2,0)$ and $(0,2)$ satisfy the circle and lie on the axes.
Equation of line through them: $\frac{x}{2} + \frac{y}{2} = 1 \Rightarrow x + y = 2$

Answer 2

Step 1: Understand the problem
A line \(L\) intersects the circle \(x^2 + y^2 = 4\) at two points. The lines joining the origin \((0,0)\) to these points of intersection coincide with the coordinate axes \(x=0\) and \(y=0\). We are to find the equation of the line \(L\).

Step 2: Parametrize the intersection points
Let the points of intersection be \(P\) and \(Q\). The lines \(OP\) and \(OQ\) are the coordinate axes, so:
- One intersection point lies on the \(x\)-axis: \(P = (a,0)\)
- The other lies on the \(y\)-axis: \(Q = (0,b)\)

Since both points lie on the circle \(x^2 + y^2 = 4\), we have:
\[ a^2 + 0 = 4 \implies a = \pm 2 \] \[ 0 + b^2 = 4 \implies b = \pm 2 \]

Step 3: Find the equation of line \(L\) passing through points \(P\) and \(Q\)
Using points \(P(2,0)\) and \(Q(0,2)\), the line equation is:
\[ \frac{x}{2} + \frac{y}{2} = 1 \implies x + y = 2 \]

Step 4: Verify
This line intersects the circle at \((2,0)\) and \((0,2)\), and the lines joining origin to these points are indeed the coordinate axes.

Final answer:
\[ \boxed{x + y = 2} \]