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

Given equation of conic is y = x²
dy/dx = 2x
Slope of tangent at (2, 10) is 4.
Equation of tangent at (2, 10) is
y - 10 = 4(x - 2)
y - 10 = 4x - 8
4x - y - 2 = 0
The tangent touches the circle x² + y² + 8x - 6y = k
The equation of the circle is x² + y² + 8x - 6y - k = 0
The distance from the center (-4, 3) to the tangent 4x - y - 2 = 0 is the radius.
Radius = |4(-4) - 3 - 2| / √(4² + (-1)²) = |-16 - 5| / √17 = 21/√17
Equation of the circle is (x + 4)² + (y - 3)² = (21/√17)²
x² + 8x + 16 + y² - 6y + 9 = 441/17
x² + y² + 8x - 6y + 25 - 441/17 = 0
x² + y² + 8x - 6y + 425/17 - 441/17 = 0
x² + y² + 8x - 6y - 16/17 = 0
Comparing this with x² + y² + 8x - 6y = k, we have k = -16/17
The equation of the tangent is 4x - y = 2
Solving this with the equation of the circle x² + y² + 8x - 6y = -16/17, we get
x = -2, y = -10
Therefore, the point of contact is (-2, -10)
However, this point is not in the options.
Let's use another approach.
The equation of the tangent at (2,10) to y=x² is given by
y - 10 = 4(x-2) => 4x - y -2 = 0
The distance from center (-4,3) of the circle to the tangent is the radius
Radius = |-16 -3 -2|/√(16+1) = 21/√17
The equation of the circle is (x+4)² + (y-3)² = (21/√17)²
Solving the equations 4x-y-2=0 and (x+4)²+(y-3)²=(21/√17)² simultaneously gives the point of contact.
4x-y=2 => y = 4x-2
(x+4)² + (4x-2-3)² = 441/17
(x+4)² + (4x-5)² = 441/17
17x² + 34x + 41 = 441/17
289x² + 578x + 697 - 441 = 0
289x² + 578x + 256 = 0
(17x+16)² = 0 => x = -16/17
y = 4(-16/17) -2 = -64/17 -34/17 = -98/17
(-16/17, -98/17) is not in options. There must be a mistake in the problem statement or options.