If m is the slope of a common tangent to the curves x²/16 + y²/9 = 1 and x² + y² = 12, then 12m² is equal to :

Solution

Maths sir · 4 steps

Write the tangent condition for each curve in slope form, then make the two intercepts equal.

1. Ellipse tangent. For x²/a²+y²/b²=1, the line y = mx + c touches the ellipse when c² = a²m² + b². Here a² = 16 and b² = 9, so c² = 16m² + 9.
2. Circle tangent. For x² + y² = r², the line y = mx + c touches the circle when c² = r²(1+m²). Here r² = 12, so c² = 12(1+m²).
3. Same line. A common tangent is one line, so its c² must satisfy both conditions at once. Set the two expressions equal.

   16m² + 9 = 12 + 12m² ⇒ 4m² = 3 ⇒ m² = 3/4

4. So. 12m² = 12 × 3/4 = 9, which is option (b).

The idea to keep. For a common tangent to two conics, write each tangency condition as c² in terms of m and equate them. One equation in m falls out.

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