Let the equation of the circle, which touches x-axis at the point (a, 0), a > 0$a > 0$ and cuts off an intercept of length b on y-axis be x^2 + y^2 - alpha x + beta y + gamma = 0$x^2 + y^2 - \alpha x + \beta y + \gamma = 0$. If the circle lies below x-axis, then the ordered pair (2a, b^2)$(2a, b^2)$ is equal to: (1) (alpha, beta^2 + 4gamma)$(alpha, \beta^2 + 4gamma)$ (2) (gamma, beta^2 - 4alpha)$(gamma, \beta^2 - 4alpha)$ (3) (gamma, beta^2 + 4alpha)$(gamma, \beta^2 + 4alpha)$ (4) (alpha, beta^2 - 4gamma)$(alpha, \beta^2 - 4gamma)$

### Related Formula Circle intercepts standard form templates: texty-intercept = 2sqrtf^2 - c$$\text{y-intercept} = 2\sqrt{f^2 - c}$$ ### Core Logic Since the circle touches the x-axis at (a,0)$(a,0)$ and lies entirely below it, its center is located at (a, -p)$(a, -p)$ where p$p$ matches its radius r$r$. By Pythagoras' theorem:

r^2 = a^2 + fracb^24 = p^2$$r^2 = a^2 + \frac{b^2}{4} = p^2$$ ### Step 1: Translating to General Equation Form The explicit standard equation is (x-a)^2 + (y+p)^2 = r^2$(x-a)^2 + (y+p)^2 = r^2$. Expanding it out: x^2 + y^2 - 2ax + 2py + a^2 = 0$$x^2 + y^2 - 2ax + 2py + a^2 = 0$$ Comparing this directly to x^2 + y^2 - alpha x + beta y + gamma = 0$x^2 + y^2 - \alpha x + \beta y + \gamma = 0$ yields: alpha = 2a$\alpha = 2a$, beta = 2p$\beta = 2p$, and gamma = a^2$\gamma = a^2$. ### Step 2: Evaluating the Target Mapped Ordered Pair Isolating b^2$b^2$ using the parametric radius dimensions: b^2 = 4p^2 - 4a^2 = (2p)^2 - 4(a^2) = beta^2 - 4gamma$$b^2 = 4p^2 - 4a^2 = (2p)^2 - 4(a^2) = \beta^2 - 4\gamma$$ Thus, the mapped ordered pair (2a, b^2)$(2a, b^2)$ evaluates directly to (alpha, beta^2 - 4gamma)$(\alpha, \beta^2 - 4\gamma)$. ### Pattern Recognition Tangency conditions fix center parameters to match radius scale sizes instantly, reducing variable overhead in coordinate transformations. ### Evaluation Rubric / Model Answer null ### Chapter Mix Class 11 Maths: Circles