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← IIT-JEE 2010 · Paper 1
MathematicsMultiple correct+3Question 37 of 84

IIT-JEE 2010 · Paper 1, Mathematics question 37

Let A and B be two distinct points on the parabola y2=4xy^2=4x. If the axis of the parabola touches a circle of radius r having AB as its diameter, then the slope of the line joining A and B can be

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Answer and worked solution

Official answer: C, D

Points. A(t12,2t1)A(t_1^2,2t_1), B(t22,2t2)B(t_2^2,2t_2) on y2=4xy^2=4x. The circle on AB as diameter has centre (t12+t222, t1+t2)\left(\frac{t_1^2+t_2^2}2,\ t_1+t_2\right).

Touching the axis (y=0y=0): the radius equals the distance of the centre from the xx-axis, r=∣t1+t2∣r=|t_1+t_2|.

Slope of AB. 2(t2−t1)t22−t12=2t1+t2=±2r\dfrac{2(t_2-t_1)}{t_2^2-t_1^2}=\dfrac2{t_1+t_2}=\pm\dfrac2r.

(C), (D)

Topics:ParabolaEquation of a circle
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