Graphing Prevents Overthinking
There are four lines that are tangent to both circles
x
2
+
y
2
=
1
and
(
x
−
6
)
2
+
y
2
=
4
.
What is the sum of the slopes of these four lines?
The answer is 0.
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2 solutions
Exactly how I solved it.
That is a good logic. But i uploaded a mathematical solution.
The combined equation of direct common tangents comes out to be x 2 − 3 5 y 2 + 1 2 x + 3 6 = 0 and that of transverse common tangents comes out to be x 2 − 3 y 2 − 4 x + 4 = 0 . The sum of slopes of DCT and TCT is 0 because there is no term of x y in their combined equations. Hence, the answer is 0 .
Looks like you go CatalyseR
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Yes, this problem appeared in CatalyseR's test series.
Is there any direct formula for finding combined equation of DCT and TCT?
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No, there's no direct formula. You have to find them by method. If you want i can upload the method.
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It seems quite obvious that graph would be symmetric about x-axis so sum of slopes would be 0.
Yes please upload the method for finding DCT and TCT
The center of both circles lie on the x-axis, so the slopes of the two tangents that are common to both circles are the negation of each other.