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Thenumber of common tangents to the circles `"x"^("2")"+y"^("2")"-4x-6y-12=0"`and `x^2+""y^2+""6x""+""18 y""+""26""=""0`,is :(1)1 (2) 2 (3) 3 (4) 4A. 1B. 2C. 3D. 4 |
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Answer» Correct Answer - C PLAN Number of common tangents depend on the position of the circle with respect to each other. (i) If circles touch internally `rArr C_(1)C_(2)=r_(1)+r_(2)`, 3 common tangents (ii) If circles touch internally `rArr C_(1)C_(2)=r_(1)+r_(2)`, 1 common tangent. (iii) If circles do not touch each other, 4 common tangents. Given equations of circles are `x^(2)+y^(2)-4x -6y-12=0" "...(i)` `x^(2)+y^(2)+6x+18y + 26=0" " ...(ii)` Centre of circle (i) is `C_(1)(2, 3)` and radius `=sqrt(4+9+12)=5(r_(1))" "["say"]` Centre of circle (ii) is `C_(2)(-3, -9)` and radius `=sqrt(9+81-26)=8(r_(2))" " ["say"]` Now, `C_(1)C_(2)=sqrt((2+3)^(2)+(3+9)^(2))` `rArr C_(1)C_(2)=sqrt(5^(2)+12^(2))` `rArrC_(1)C_(2)=sqrt(25+144)=13` `thereforer_(1)+r_(2)=5+8+13` Also `C_(1)C_(2)=r_(1)+r_(2)` Thus, both circles touch each other externally, Hence, there are three common tangents. |
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