2.6.9 Circle proof

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Fixed question: Given two points at opposite ends of a diameter, write down the equation of the circle.

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Merryn Horrocks

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Merryn Horrocks 11 months ago

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Question statement

Show that if $A(x_1, y_1)$ and $B(x_2, y_2)$ are at opposite ends of a diameter of a circle then the equation of the circle is $(x − x_1)(x − x_2) + (y − y_1)(y − y_2) = 0$ .

Hint: if $P$ is any point on the circle, obtain the slopes of the lines $AP$ and $BP$ and recall that the angle in a semicircle must be a right angle.

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Let $P (x,y)$ be an arbitrary point on the circle.

The the angle $\angle APB$ is a right angle, meaning that $AP$ and $BP$ are perpendicular, meaning that the product of their gradients is $-1$ .

Let $m_1$ be the gradient of $AP$ and let $m_2$ be the gradient of $BP$ .

Then

$m_1 = \dfrac{y-y_1}{x-x_1}\quad$ and $\quad m_2=\dfrac{y-y_2}{x-x_2}$

$m_1 m_2 = -1$

$\dfrac{y-y_1}{x-x_1} \times \dfrac{y-y_2}{x-x_2}=-1$

Multiplying through by the denominator:

$(y-y_1)(y-y_2) = -(x-x_1)(x-x_2)$

i.e.

$(x-x_1)(x-x_2)+(y-y_1)(y-y_2)=0$

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This question is used in the following exams:

HELM Book 2.6 The Circle by Merryn Horrocks in HELM books.
HELM Book 2.6.2 Exercises by Merryn Horrocks in HELM books.

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Merryn Horrocks 11 months ago

Merryn Horrocks 2 years, 4 months ago

Merryn Horrocks 2 years, 4 months ago

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