Let A and B be two distinct points on the parabola . If the axis of the parabola touches a circle of radius having AB as its diameter, then the slope of the line joining A and B can be [2010]

Question

Let A and B be two distinct points on the parabola $y^{2} = 4 x$ . 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 [2010]

Smart Achievers · Accepted Answer

(3, 4)

Given equation of parabola is $y^{2} = 4 x$

Its axis is $x$ -axis.

Let $A (t_{1}^{2}, 2 t_{1})$ and $B (t_{2}^{2}, 2 t_{2})$

Then centre of circle drawn with AB as diameter is

$(\frac{t_{1}^{2} + t_{2}^{2}}{2}, t_{1} + t_{2})$

As circle touches axis of parabola i.e., $x$ -axis

$∴$ $r =$ $| t_{1} + t_{2} |$ $\Rightarrow t_{1} + t_{2} = \pm r$

$∴$ $Slope of A B = \frac{2 (t_{2} - t_{1})}{t_{2}^{2} - t_{1}^{2}} = \frac{2}{t_{2} + t_{1}} = \pm \frac{2}{r}$

Solution

Q.
Let A and B be two distinct points on the parabola $y^{2} = 4 x$ . 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 [2010]

1 $- \frac{1}{r}$

2 $\frac{1}{r}$

3 $\frac{2}{r}$

4 $- \frac{2}{r}$

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Solution

Q. Let A and B be two distinct points on the parabola y2=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 [2010]

1 -1r

2 1r

3 2r

4 -2r