[Math] Parabola and Circle problem : The parabola $y =x^2-8x+15$ cuts the x axis at P and Q. A circle is drawn ……….

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Problem :

The parabola $y=x^2-8x+15$ cuts the x axis at P and Q. A circle is drawn through P and Q so that the origin is outside it. Find the length at a tangent to the circle from O.

My approach :

Since the parabola $y=x^2-8x+15$ cuts the x axis therefore, its y coordinate is zero,

Solving the equation: $x^2-8x+15=0$ we get two points $(3,0)$ and $(5,0)$.

Now how to proceed further with these two points, please suggest. thanks..

Best Answer

Where is the centre of the circle - at some point

$C=(4,a)$

What is the square of the radius of the circle:

$r^2=a^2+1$

What is the square of the distance from the origin to the centre of the circle:

$OC^2=4^2+a^2$

Let $S$ be a point on the circle where the tangent from the origin touches it. We have a right-angled triangle with $OS^2+CS^2=OC^2$ and we know that $CS^2=r^2$

Can you finish it from there?

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