Circle And System Of Circles Question 28
Question: The number of common tangents to the circles $ x^{2}+y^{2}-x=0,,x^{2}+y^{2}+x=0 $ is
[EAMCET 1994]
Options:
A) Length of AB is constant
B) PA and PB are not always equal
C) The locus of the midpoint of AB is $ x^{2}+y^{2}=2x^{2}+2y^{2} $
D) None of these
Show Answer
Answer:
Correct Answer: C
Solution:
Let $ P(x_1,\ y_1) $ be a point on $ x^{2}+y^{2}=4 $ .
Then the equation of the tangent at P is $ xx_1+yy_1=4 $ which meets the coordinate axes at $ A( \frac{4}{x_1},\ 0 ) $ and $ B( 0,\ \frac{4}{y_1} ) $ .
Obviously, (a) and (b) are not true.
Let $ (h,\ k) $ be the mid-point of AB. Therefore $ h=\frac{x_1 + x_2}{2},\ k=\frac{y_1 + y_2}{2} $ i.e., $ x_1=\frac{2h - x_2}{1},\ y_1=\frac{2k - y_2}{1} $
But $ (x_1,\ y_1) $ lies on $ x^{2}+y^{2}=4 $ .
BETA
