Mathematics - Circle Question with Solution | TestHub

Here's a breakdown of the process and what "squaring we get" implies:

1. Identify Given Information:

* Center: Let the center be $(h,k)$ (sometimes $(x,y)$ for the locus).

* Touches y-axis: Radius $r=|h|$ (distance from center to y-axis).

* Touches another circle externally: Distance between centers = Sum of radii.

2. Set up the Distance Equation:

* Let the given circle have center $(C_x,C_y)$ and radius $R_{given}$.

* Distance between centers $(h,k)$ and $(C_x,C_y)$ is $\sqrt{(h-C_x{)}^2+(k-C_y{)}^2}$.

* Sum of radii is $r+R_{given}=|h|+R_{given}$.

* So, $\sqrt{(h-C_x{)}^2+(k-C_y{)}^2}=|h|+R_{given}$.

3. "Squaring we get": This step removes the square root to simplify the equation, leading to the locus.

4. * Squaring both sides: $(h-C_x{)}^2+(k-C_y{)}^2=(|h|+R_{given}{)}^2$.

* Expanding this gives: $h^2-2h{C}_x+C_x^2+k^2-2k{C}_y+C_y^2=h^2+2|h|R_{given}+R_{given}^2$ (since $|h{|}^2=h^2$).

* Simplifying by canceling $h^2$: $-2h{C}_x+C_x^2+k^2-2k{C}_y+C_y^2=2|h|R_{given}+R_{given}^2$.

5. Final Locus Equation: The resulting equation (after moving terms and potentially handling the absolute value based on the quadrant) will define the locus, often a parabola if it touches the x-axis too, or a hyperbola if touching two fixed circles (as mentioned in), depending on the exact setup.