Let tangents are drawn at two points of the circle (x−7)2+(y+1)2=25. If the point of intersection of both the tangents is origin, then the angle between them (in degrees) is
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Solution
Any line passing through origin is written as y−mx=0
This line is tangent to the circle (x−7)2+(y+1)2=25, so the distance from centre to the line is equal to radius, |−1−7m|√1+m2=5
Squaring on both the sides, we get (1+7m)2=25(1+m2) ⇒24m2+14m−24=0D=142+4×242>0
Let the roots of the equation be m1,m2, so m1m2=−2424=−1
Hence, the angle between the tangents is 90∘
Alternate solution :
Given circle is (x−7)2+(y+1)2=25
Distance between OC =√72+12=√50=5√2
From the figure, we get sinθ2=CPOC⇒sinθ2=55√2=1√2⇒θ2=45∘∴θ=90∘