Find the equation of circle with radius r, whose centre lies in 1st qu

1.	Find the equation of circle with radius r, whose centre lies in 1st quadrant and touches y-axis at a distance of h from the origin. Find the equation of other tangent which passes through origin.
Answer» Centre of circle = (r, h) Radius of circle = r Thus, equation of circle (x – r)² + (y – h)² = r² Let tangent OB touches the circle at B. Tangent passes through origin. Let tangent touches the circle at point (x₁,y₁). ∴ Equation of tangent at point of circle, xx₁ + yy₁ + g(x + x₁) + f(y + y₁) + c = 0 xx₁ + yy₁ – r(x + x₁) + h(y + y₁) + h² = 0 Since the line passes through origin. ∴ x × 0 + y × 0 – r(x + 0) – h(y + 0) + h² = 0 ⇒ – rx – hy + h² = 0 ⇒ rx + hy – h² = 0 This is required equation.

Find the equation of circle with radius r, whose centre lies in 1st quadrant and touches y-axis at a distance of h from the origin. Find the equation of other tangent which passes through origin.

Answer»

Centre of circle = (r, h)

Radius of circle = r

Thus, equation of circle

(x – r)² + (y – h)² = r²

Let tangent OB touches the circle at B.

Tangent passes through origin.

Let tangent touches the circle at point (x₁,y₁).

∴ Equation of tangent at point of circle,

xx₁ + yy₁ + g(x + x₁) + f(y + y₁) + c = 0

xx₁ + yy₁ – r(x + x₁) + h(y + y₁) + h² = 0

Since the line passes through origin.

∴ x × 0 + y × 0 – r(x + 0) – h(y + 0) + h² = 0

⇒ – rx – hy + h² = 0

⇒ rx + hy – h² = 0

This is required equation.