Question

If the tangent to the conic, y – 6 = x² at (2, 10) touches the circle, x² + y² + 8x – 2y = k (for some fixed k) at a point (α, β); then (α, β) is ______.

Options

• `(-7/17, 6/17)`

• `(-4/17, 1/17)`

• `(-6/17, 10/17)`

• `(-8/17, 2/17)`

Answer 1

If the tangent to the conic, y – 6 = x² at (2, 10) touches the circle, x² + y² + 8x – 2y = k (for some fixed k) at a point (α, β); then (α, β) is `underlinebb((-8/17, 2/17)`.

Explanation:

x² – y + 6 = 0

`\implies 2x - (dy)/(dx)` = 0

`\implies (dy)/(dx)` = 2x

`dy/dx|_((x","y) = (2"," 10))` = 4

Equation of tangent

y – 10 = 4(x – 2)

`\implies` 4x – y + 2 = 0

Tangent passes through (α, β)

4α – β + 2 = 0

`\implies` β = 4α + 2  ...(i)

and 2x + 2yy' + 8 – 2y' = 0

y' = `(2x + 8)/(2 - 2y) = (2α + 8)/(2 - 2β)` = 4  ...(ii)

From (i) and (ii)

α = `(-8)/17`, β = `2/17; (-8/17, 2/17)`