If and are two common tangents of circle and parabola , then the value of is equal to

Question

If  and  are two common tangents of circle  and parabola , then the value of  is equal to

Canvas Classes · Accepted Answer

🧠 Common Tangents to Circle and Parabola — Slopes Come from a Single Equation

Read the question. Common tangents to the circle  (radius , centred at origin) and parabola  (). Find .

Parabola's slope-form tangent: , or .

Distance from origin to this line equals  (circle's radius):

.

Square: , so , hence .

Let . Then , giving .

. So .

(The other root is negative, ruled out since .)

So . The two real values of  are , giving slopes  and .

Product: , so .

.

🗺️ Working It Out

Tangent to

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