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,

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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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