Let L be a common tangent line to the curves
4x² + 9y² = 36 and (2x)² + (2y)² = 31. Then the
square of the slope of the line L is __________.

Solution

Maths sir · 4 steps

Write the tangent condition for each curve with the same slope m, then set the two intercepts equal.

1. Standard forms. Divide the first curve by 36 to get the ellipse x²/9+y²/4=1, so a²=9 and b²=4. The second curve is 4x²+4y²=31, which is the circle x²+y²=31/4.
2. Tangent conditions. The line y=mx+c touches the ellipse when c²=a²m²+b²=9m²+4. It touches the circle when c²=r²(1+m²)=31/4(1+m²).

   9m²+4=31/4(1+m²)

3. Solve for m squared. Multiply both sides by 4: 36m²+16=31+31m². This gives 5m²=15.
4. So. m²=3, so the square of the slope of L is 3.

The idea to keep. For a common tangent, write y=mx+c, use the condition c²=… for each curve, and equate the two values of c². The slope comes out directly.

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