Question

The equation of the line joining the origin to the points of intersection of the curve $x^2+y^2=a^2$and $x^2+y^2-ax-ay=0$ is:

• A. $x^2-y^2=0$
• B. $xy=0$
• C. $xy-x^2=0$
• D. $y^2+xy=0$

Answer 1

The correct option is B

$xy=0$

Determine the equation of the line :

Given that: $x^2+y^2=a^2$ and $x^2+y^2-ax-ay=0$ intersects.

This means they have a point of intersection.

Finding the point of intersection of $x^2+y^2=a^2$ and $x^2+y^2-ax-ay=0$:

$$\begin{gathered} x^2+y^2-ax-ay=0 \\ \Rightarrow x^2+y^2=a(x+y) \\ \Rightarrow {\left(x^2+y^2\right)}^2={\left[a(x+y)\right]}^2 \\ \Rightarrow {\left(x^2+y^2\right)}^2=a^2{(x+y)}^2 \\ \Rightarrow {\left(x^2+y^2\right)}^2=(x^2+y^2){(x+y)}^2 \\ \Rightarrow x^2+y^2={\left(x+y\right)}^2 \\ \Rightarrow x^2+y^2=x^2+y^2+2xy \\ \Rightarrow 2xy=0 \\ \Rightarrow xy=0 \end{gathered}$$

Therefore, the equation of the required line is $xy=0.$ It satisfies the origin coordinates as well.

Hence, the correct answer is Option (B).