If a circle passing through the point (−4,3) also touches the lines x+y=2 and x−y=2, then which of the following is/are true?
A
The centre of the circle is (−10+3√6,0)
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B
The centre of the circle is (−10−3√6,0)
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C
The radius of the circle is r=6√2−3√3 units
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D
The radius of the circle is r=6√2+3√3 units
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Solution
The correct option is D The radius of the circle is r=6√2+3√3 units Given lines are x+y=2 and x−y=2 which are perpendicular in nature,
We know that, r=∣∣∣h−k−2√2∣∣∣=∣∣∣h+k−2√2∣∣∣=√(h+4)2+(k−3)2
Now, using the above conditions ∣∣∣h−k−2√2∣∣∣=∣∣∣h+k−2√2∣∣∣
By observation, point (−4,3) and centre of the circle lies on the same side of both the lines, so −(h−k−2√2)=−(h+k−2√2)⇒k=0
Now, ∣∣∣h+k−2√2∣∣∣=√(h+4)2+(k−3)2⇒∣∣∣h−2√2∣∣∣=√(h+4)2+9
Squaring on both sides, we get ⇒(h−2)2=2(h2+8h+25)⇒h2+20h+46=0⇒h=−20±√400−1842⇒h=−10±3√6
So, the centre of the circle are (−10+3√6,0) and (−10−3√6,0)
Therefore, r=|−10+3√6−2|√2 and r=|−10−3√6−2|√2⇒r=|−6√2+3√3| and r=|−6√2−3√3|∴r=6√2−3√3 and r=6√2+3√3