The line 12x cos θ + 5y sin θ = 60 is tangent to which of the following curves?

Solution

Maths sir · 4 steps

Divide by 60 and the line turns into the standard parametric tangent of an ellipse. Read off a and b.

1. Divide by 60. Divide both sides by 60. Since 12/60 = 1/5 and 5/60 = 1/12, the line becomes the equation below.

   (x cos θ)/5 + (y sin θ)/12 = 1

2. Match the form. The tangent to x²/a² + y²/b² = 1 at the point (a cos θ, b sin θ) is (x cos θ)/a + (y sin θ)/b = 1. Here a = 5 and b = 12, so the curve is x²/25 + y²/144 = 1.
3. Clear fractions. Multiply by 25 × 144 = 3600 to get 144x² + 25y² = 3600. As a check, the condition a²l² + b²m² = n² gives 25(144 cos² θ) + 144(25 sin² θ) = 3600 = 60² for every θ.
4. So. The line touches the ellipse 144x² + 25y² = 3600 for every value of θ, which is option (b).

The idea to keep. When a line contains cos θ and sin θ, make the right side 1. If it looks like (x cos θ)/a + (y sin θ)/b = 1, it is the tangent to x²/a² + y²/b² = 1.

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