(a)

The equation of parabola is  and the point is .

Slope of the tangent is derivative of the curve.

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Apply derivative on each side with respect to .

Slope of the tangent  is .

Point-slope form of line equation : .

Substitute and in the above formula.

This is a pair of tangent lines.

These tangent lines intersect the parabola, and the intersecting points can be determined  by solving them.

Substitute in the curve .

and .

Substitute  values in .

If , then .

If , then .

Therefore, the points at tangent lines intersect parabola are and .

Tangent line passing through :

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Find the slope at .

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Point-slope form of line equation : .

Substitute and in the above formula.

Tangent line  passing through :

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Find the slope at .

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Point-slope form of line equation : .

Substitute and in the above formula.

(b)

The equation of parabola is  and the point is .

Slope of the tangent to parabola is .

At , .

Assume that at is the tangent point.

Slope of the tangent line at is .

Slope of the line passing through two points and is defined as .

Here and

Discriminant of quadratic equation is .

Here and .

Since the discriminant is negative there is no real values of .

Therefore there is no tangent line at .

Graph:

Graph the curve with the point .

Tangents are and .

Graph is

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