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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 y−axis. Let the other common tangent has slope m. Then, its equation is y=mx+1m or, mx−y+1m=0 Perpendicular distance from the centre (−1,0) to the tangent mx−y+1m=0 is equal to the radius. ⇒∣∣ ∣ ∣∣−m+1m√m2+1∣∣ ∣ ∣∣=1 ⇒(m−1m)2=m2+1 ⇒1m2=3 ⇒m=±1√3 i.e., ∠BAO=∠OAC=π6 ⇒∠BAC=π3 and AB=AC Hence, the triangle ABC is equilateral.

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