The tangent to the parabola at a point touches the circle at a point Q. What are the coordinates of Q?

# The tangent to the parabola at a point touches the circle at a point Q. What are the coordinates of Q?

1. A
2. B
3. C
4. D

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### Solution:

Let the equation of the line using the point-slope formula be … (1)
To find the slope of the line, differentiate the given equation with respect to x, we get
We know that, at point , we get
To find the equation of the tangent, substitute the values and in (1), we get
… (2)
Now, the equation of the circle is . Let the standard form of the equation of the circle be with centre .
Therefore, the centre of the circle is . So, the tangent touches the parabola at the point P, and also touches the circle with the centre O at the point Q.
To find the radius of the circle, substitute the point and the tangent line , in the distance formula. We know that, .
Hence, the radius of the circle is .
Using the trial-and-error method, check the options, the distance between the point Q and O is r.
So,
Hence, the correct option is 3.

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