| Statement I The number of common tangents to the circle x2 + y2 = 4 and x2 + y2 - 8x - 6y - 24 = 0 is 4. |
| Statement II Circle with centre \[{{c}_{1}},{{c}_{2}}\] and radii \[{{r}_{1}},{{r}_{2}}\] and if \[|{{e}_{1}}{{e}_{2}}|>{{r}_{1}}+{{r}_{2}}\] , then circles have 4 common tangents. |
A) Both Statement I and Statement II are true and the Statement II is the correct explanation of the Statement I
B) Both Statement I and Statement II are true but the Statement II is not the correct explanation of the Statement I
C) Statement I is true but Statement II is false
D) Statement I is false but Statement II is true
Correct Answer: D
Solution :
Idea Two circles with centre \[{{c}_{1}}({{x}_{1}},{{y}_{1}})\] and \[{{c}_{2}}({{x}_{2}},{{y}_{2}})\] radii \[{{r}_{1}}\] and \[{{r}_{2}}\] respectively, touch each other internally if\[{{c}_{1}}{{c}_{2}}={{r}_{1}}-{{r}_{2}}\] and we know that, when circle, circle touch each other internally. Only one common tangent can be drawn to the circles. The given circle is x2 + y2 = 4 Therefore, \[{{c}_{1}}(0,0),{{r}_{1}}=2\] and x2 + y2 - 8x - 6y - 24 = 0 \[\therefore \] \[{{c}_{2}}=(4,3),{{r}_{2}}=7\] \[\therefore \] \[{{c}_{1}}{{c}_{2}}=5\]and\[|{{r}_{2}}-{{r}_{1}}|=5\] Now, \[{{c}_{1}}{{c}_{2}}=|{{r}_{2}}-{{r}_{1}}|\] So, it is clear that circles touch internally. \[\therefore \]Only one tangent is possible, \[\therefore \]Statement I is false but Statement II is true. TEST Edge External contact of circles with common tangent to two circles based questions are asked. To solve such type of question, students are advised to understand the concept of circle.
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