Circles Question - None

Question

Two circles are given by $$x^2 + y^2 - 8 = 0$$ and $$(x - a)^2 + y^2 - 8 = 0$$. The common chord is obtained by subtracting the equations: $$2ax - a^2 = 0$$, which gives $$x = \frac{a}{2}$$. If the common chord subtends a right angle at the origin, find the value of a.

Accepted Answer

Step 1: The common chord is: $$2ax - a^2 = 0 \Rightarrow x = \frac{a}{2}$$

Step 2: Substituting in $$x^2 + y^2 = 8$$:

$$\frac{a^2}{4} + y^2 = 8$$

$$y^2 = 8 - \frac{a^2}{4}$$

Step 3: The endpoints of the chord are $$\left(\frac{a}{2}, \pm\sqrt{8 - \frac{a^2}{4}}\right)$$

Step 4: For the chord to subtend a right angle at origin O, if the endpoints are P and Q, then $$\vec{OP} \cdot \vec{OQ} = 0$$

Step 5: $$\frac{a^2}{4} - \left(8 - \frac{a^2}{4}\right) = 0$$

$$\frac{a^2}{2} = 8$$

$$a^2 = 16$$

$$a = 4$$

Question:

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