If the line y = mx + c is a common tangent to the hyperbola
x²/100 - y²/64 = 1 and the circle x² + y² = 36, then which one of the following is true?

Solution

Maths sir · 4 steps

Write the tangency condition for each curve, set the two values of c² equal, and solve for m.

1. Hyperbola condition. For x²/a² - y²/b² = 1, the line y = mx + c is a tangent when c² = a²m² - b². Here a² = 100 and b² = 64, so c² = 100m² - 64.
2. Circle condition. For x² + y² = r², the line is a tangent when c² = r²(1 + m²). With r² = 36, this gives c² = 36 + 36m².

   100m² - 64 = 36m² + 36 ⇒ 64m² = 100 ⇒ m² = 25/16

3. Find c squared. So m = ±5/4, and c² = 36(1 + 25/16) = 36 × 41/16 = 369/4. Check with the hyperbola: 100 × 25/16 - 64 = 369/4 as well.
4. So. c² = 369/4, which means 4c² = 369. The answer is (d).

The idea to keep. For a common tangent, write c² from each curve's tangency condition and equate them. One equation in m falls out, and c² follows.

Answer from the NTA final answer key. Solution by PrepWiser.