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(D) Since the circle passes through the origin $(0, 0)$ and cuts intercepts of length $3$ and $4$ on the positive $x$ and $y$ axes respectively,it pas

Vedclass · Accepted Answer

$x^2 + y^2 - 3x - 4y = 0$

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(D) Since the circle passes through the origin $(0, 0)$ and cuts intercepts of length $3$ and $4$ on the positive $x$ and $y$ axes respectively,it passes through the points $(3, 0)$ and $(0, 4)$.The general equation of a circle passing through the origin is $x^2 + y^2 + 2gx + 2fy = 0$.Substituting the point $(3, 0)$ into the equation:$3^2 + 0^2 + 2g(3) + 2f(0) = 0$ $\Rightarrow 9 + 6g = 0$ $\Rightarrow g = -\frac{3}{2}$.Substituting the point $(0, 4)$ into the equation:$0^2 + 4^2 + 2g(0) + 2f(4) = 0$ $\Rightarrow 16 + 8f = 0$ $\Rightarrow f = -2$.Substituting $g$ and $f$ back into the general equation:$x^2 + y^2 + 2(-\frac{3}{2})x + 2(-2)y = 0$$x^2 + y^2 - 3x - 4y = 0$.

The equation of the circle passing through the origin and cutting intercepts of length $3$ and $4$ units from the positive axes is:

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