Question

A variable circle passes through the point $A(2,1)$ and touches the x-axis. Locus of the other end of the diameter through $A$ is a parabola.
(i) Find the length of the rectum of the parabola.

Answer 1

We have,

A variable circle passes through the point $A(2,1)$and touches the X-axis.

So,

In a circle AB is a diameter where the coordinate of point A is (2,1) and let the coordinate of B is $(x_1,y_1)$.

Now, we know that,

Equation of circle in diameter from is

$(x-x_1)(x-2)+(y-y_1)(y-1)=0$

And it touches X-axis so $y=0$

$(x-x_1)(x-2)+(0-y_1)(0-1)=0$

$\Rightarrow x^2-x{x}_1-2x+2x_1+y_1=0$

$\Rightarrow x^2+(-2-x_1)x+2x_1+y_1=0$

Also, the discriminate of above equation will be equal to zero,

Because circle touches X-axis.

${(-2-x_1)}^2=4(2x_1+y_1)$

$\Rightarrow 4+4x_1+{x_1}^2=8x_1+4y_1$

$\Rightarrow 4-4x_1+{x_1}^2=4{y_1}^2$

$\Rightarrow {(x_1-2)}^2=4{y_1}^2$

Hence, the locus of end pint of diameter.

This is the answer.