$$Circle(s)$$ touching $$x-axis$$ at a distance $$3$$ from the origin and having an intercept of length $$27$$ on $$y-axis$$, is (are)

Question

$$Circle(s)$$ touching $$x-axis$$ at a distance $$3$$ from the origin and having an intercept of length $$27$$ on $$y-axis$$, is (are)

Infinity Learn · Accepted Answer

Let the equation of the circle be $$x2+y2+2gx+2fy+c=0$$. It touches $$x-axis$$ at a distance $$3$$ from the origin. Therefore, $$c=g2$$ and the circle passes through $$($$±$$3$$, $$0)$$.∴    $$9$$±$$6g+c=0$$⇒    $$9$$±$$6g+g2=0$$⇒$$(g$$±$$3)2=0$$⇒$$g=$$−$$3$$∴    $$c=g2$$⇒$$c=9The$$ circle cuts an intercept of length $$27$$ on $$y-axis$$.∴ $$2f2$$−$$c=27$$⇒$$f2$$−$$9=7$$⇒$$f=$$±$$4Hence$$, the equations of the circles are   $$x2+y2$$−$$6x$$±$$8y+9=0$$

Circle $(s)$ touching $x -$ axis at a distance $3$ from the origin and having an intercept of length $2 \sqrt{7}$ on $y -$ axis, is (are)

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Circle(s) touching x-axis at a distance 3 from the origin and having an intercept of length 27 on y-axis, is (are)

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Circle $(s)$ touching $x -$ axis at a distance $3$ from the origin and having an intercept of length $2 \sqrt{7}$ on $y -$ axis, is (are)