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Question
If angle between two tangents drawn from a point P to a circle of radius a and centre O is 90°, then OP = `asqrt(2)`.
Options
True
False
Solution
This statement is True.
Explanation:
Tangent is always perpendicular to the radius at the point of contact.
Hence, ∠RPT = 90°
If 2 tangents are drawn from an external point, then they are equally inclined to the line segment joining the centre to that point.
Consider the following figure,
From point P, two tangents are drawn.
It is given that, OT = a
And line OP bisects ∠RPT.
So,
∠TPO = ∠RPO = 45°
We know that, OT ⊥ PT
In right-angled triangle OTP,
sin 45° = `"OT"/"OP"`
= `1/sqrt(2) = "a"/"OP"`
Hence, OP = `asqrt(2)`
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