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Topic
In the coordinate plane Oxygengive points USA(1 ; 1) and straight line : 3x + 4y + 3 = 0. Write the equation of the circle (OLD), know (OLD) has a mind USA and straight line cut (OLD) at two points WOMEN, P satisfy the triangle MNP even.
Solution method – See details
Find the radius of the circle (OLD)
Step 1: Calculate the distance from M to
Step 2: ConsiderMNP Everyone knows the length of the altitude line from USAcalculate the lengths of the sides of the triangle as the radius of (OLD)
Step 3: Write a circle PT with center USA and the radius found in step 2
Detailed explanation
Call H is the projection of USA go straight
We have: \(MH = d(M,\Delta ) = \frac{{\left| {3 + 4 + 3} \right|}}{{\sqrt {{3^2} + {4^2} } }} = 2\)
According to the assumption,MNP evenly \( \Rightarrow \widehat {MNH} = {60^0}\)
Consider \(\Delta MNH\) square at H yes \(MN = \frac{{MH}}}{{\sin \widehat {MNH}}} = \frac{2}{{\sin {{60}^0}}} = \frac{{4\sqrt 3 }}{3}\)\( \Rightarrow R = \frac{{4\sqrt 3 }}{3}\)
So (OLD) has PT: \({(x – 1)^2} + {(y – 1)^2} = \frac{{16}}{3}\)
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