Math Assignment Help With Condition Of Tangency

7.5.7 Condition of tangency:

Theorem: The condition of tangency states that the line y = mx + c touches the parabola y² = 4ax at

c = a/m

Proof: let y = mx + c be the line intersecting the parabola y² = 4ax

y² = 4ax, y = mx + c

(mx + c)² = 4ax

m² x² + c² + 2mxc = 4ax

m² x² +2(mc – 2a)x + c2 = 0

The line will touch the parabola if it intersects at one point only. This will happen only when the roots are real and coincident.

For

y = mx + c being a tangent to y² = 4ax

we must have,

4(mc – 2a)² – 4m²c² = 0

4a² – 4amc = 0

Therefore for all values of m the line y = mx + a/m is always tangent to the parabola y² = 4ax

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