If the line y = mx + c is tangent to the circle x2 + y2 = 5r2 and the parabola y2 – 4x – 2y + 4λ + 1 = 0 and point of contact of the tangent with the parabola is (8, 5), then find the value of (25r2 + λ + 2m + c).

Engineering

Mathematics

A Circle

Tangent of Parabola

Line and Circle

Question

If the line y = mx + c is tangent to the circle x² + y² = 5r² and the parabola y² – 4x – 2y + 4λ + 1 = 0 and point of contact of the tangent with the parabola is (8, 5), then find the value of (25r² + λ + 2m + c).

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Parabola : (y – 1)² = 4(x – 4), λ = 4
tangent to parabola is y – 1 = m (x – 4) + $\frac{1}{m}$
it passes through (8, 5) ⇒ 4 = 4m + $\frac{1}{m}$
Hence, m = $\frac{1}{2}$
∴ equation of tangent is y = $\frac{x}{2}$ + 1 = mx + c
Hence, c = 1
Now, y = $\frac{x}{2}$ + 1 is tangent to the circle x² + y² = 5r² .
$\Rightarrow |\frac{2}{\sqrt{5}}| = \sqrt{5} r \Rightarrow r = \frac{2}{5}$
∴ 25r² + λ + 2m + c = 4 + 4 + 1 + 1 = 10.