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Question

The triangle formed by the common tangents to the parabola y2=4x and the circle x2+y2+2x=0 is

A
an equilateral triangle
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B
a right angled isosceles triangle
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C
an isosceles but not right angled triangle
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D
a scalene triangle
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Solution

The correct option is A an equilateral triangle

As circle is (x+1)2+y2=1, one of the common tangent is along the yaxis.
Let the other common tangent has slope m.
Then, its equation is y=mx+1m
or, mxy+1m=0
Perpendicular distance from the centre (1,0) to the tangent mxy+1m=0 is equal to the radius.
∣ ∣ ∣m+1mm2+1∣ ∣ ∣=1
(m1m)2=m2+1
1m2=3
m=±13
i.e., BAO=OAC=π6
BAC=π3 and AB=AC
Hence, the triangle ABC is equilateral.

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