The locus of the point of intersection of the lines x cos α + y sin α = a and x sin α - y cos α = b is

Question

Question

Answer 1

Concept:

The general equation of a non-degenerate conic section is: ax2 + 2hxy + by2 + 2gx + 2fy + c = 0 where a, h and b are all not zero

The above-given equation represents a non-degenerate conics whose nature is given below in the table:

S.No	Condition	Nature of Conic
1	h = 0 and a = b	Circle
2	h = 0 and Either a = 0 or b = 0	Parabola
3	h = 0, a ≠ b and ab > 0	Ellipse
4	h = 0, a ≠ b and sign of a and b are opposite	Hyperbola

Calculation:

(i)

(ii)

Squaring both sides of (i) and (ii) and adding both the equation we get x²+ y²= a²+ b².

By comparing the given equation with ax2 + 2hxy + by2 + 2gx + 2fy + c = 0, we get

⇒ a = 1, h = 0, b = 1

Here, we can see that h = 0, a = b

Which represents the locus of the circle.