Geometry-If-the-area-of-the-triangle-formed-by-the-positive-x-axis-the-normal-and-the-tangent-to-the-circle-x-2-y-3-25-at-the-point-5-7-is-A-then-24A-is-equal-to-

Question Number 135570 by bemath last updated on 14/Mar/21

$$Geometry$$If the area of the triangle formed by the positive x-axis, the normal and the tangent to the circle (x-2) ² +(y-3) ² =25 at the point (5,7) is A, then 24A is equal to______?

Answered by EDWIN88 last updated on 14/Mar/21

Commented by EDWIN88 last updated on 14/Mar/21

$$\mathrm{I}\mathrm{guess}\mathrm{the}\mathrm{question}\mathrm{area}\mathrm{of}\mathrm{triangle}\mathrm{formed}$$$$\mathrm{by}\mathrm{x}-\mathrm{axis},\mathrm{tangent}\mathrm{and}\mathrm{normal}\mathrm{line}$$$$\mathrm{tangent}\mathrm{line}:3\mathrm{x}+4\mathrm{y}=43\Rightarrow \mathrm{intersect}\mathrm{the}$$$$\mathrm{x}-\mathrm{axis}\mathrm{at}\mathrm{G}(\frac{43}{3},0)$$$$\mathrm{normal}\mathrm{line}:4\mathrm{x}-3\mathrm{y}=-1\Rightarrow \mathrm{intersect}\mathrm{the}$$$$\mathrm{x}-\mathrm{axis}\mathrm{at}\mathrm{P}(-\frac{1}{4},0)\mathrm{and}\mathrm{intersect}\mathrm{the}\mathrm{y}-\mathrm{axis}$$$$\mathrm{at}\mathrm{R}(0,\frac{1}{3}).$$$$\mathrm{the}\mathrm{area}\mathrm{of}\mathrm{triangle}\mathrm{is}\mathrm{A}=\frac{1}{2}.7.(\frac{43}{3}+\frac{1}{4})$$$$\mathrm{A}=\frac{7}{2}.\left(\frac{175}{12}\right)=\frac{1125}{24}.\mathrm{Therefore}$$$$24\mathrm{A}=1125$$

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