TS EAMCET · Maths · Circle

A point that lies on the common tangent to the circles $x^{2} + y^{2} - 2 x + 18 y + 78 = 0$ and $x^{2} + y^{2} + 8 x - 6 y - 200 = 0$ among the following options is

A $(0, \frac{139}{12})$
B $(\frac{- 137}{5}, \frac{- 1}{6})$
C $(31, \frac{- 4}{3})$
D $(\frac{- 2}{5}, \frac{- 47}{4})$

Verified Solution

Answer & Solution

Correct Answer

(D) $(\frac{- 2}{5}, \frac{- 47}{4})$

Step-by-step Solution

Detailed explanation

Given circles are S1≡x2+y2-2x+18y+78=0......1 Centre C1≡1,-9 & radius r1=-12+92-78=2 units And x2+y2+8x-6y-200=0......2 Centre C2=-4,3 & radius r2=42+-32+200=15 units C1C2=-4-12+3+92=13=r2-r1 ∴Circles touch each other internally. Equation of common tangent to…

Apply the relevant identity from the chapter and substitute the values established in the previous step, keeping the original constraint in view.

Use the simplified expression to isolate the unknown, then plug the result back into the question setup to confirm the working is internally consistent.

Cross-check against a boundary or limiting case. Both the dimensional balance and the qualitative behaviour at the extreme should agree with the candidate answer.

Combine the steps above to identify the correct option. The remaining choices each fail at least one of the consistency, dimensional, or boundary checks.

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A point that lies on the common tangent to the circles x2+y2-2x+18y+78=0 and x2+y2+8x-6y-200=0 among the following options is

Step-by-step Solution

A point that lies on the common tangent to the circles $x^{2} + y^{2} - 2 x + 18 y + 78 = 0$ and $x^{2} + y^{2} + 8 x - 6 y - 200 = 0$ among the following options is