Circles
Circle
Allen Star Batch
Grade 11
Question:
A variable circle passes through the origin $O$ and cuts off portions $OP$ and $OQ$ from $x$-axis and $Y$-axis respectively such that $m(OP) + n(OQ)$ is equal to unity. If the circle passes through a fixed point $(x_1, y_1)$ other than $O$, then:
$x_1^2 + y_1^2 = m^2 + n^2$
$x_1 + y_1 = m + n$
$mx_1 + ny_1 = 1$
$nx_1 - my_1 = 0$
Step-by-Step Solution
Key Concept: Intercepts on coordinate axes directly give the parameters $a$ and $b$; the linear constraint relates them.
For circle $x^2 + y^2 - ax - by = 0$ cutting axes at $P(a,0)$ and $Q(0,b)$, we have $OP = a$ and $OQ = b$. Given $m\cdot OP + n\cdot OQ = 1$, we get $ma + nb = 1$ or $b = \frac{1-ma}{n}$. This constraint determines the family of circles satisfying the given linear condition.
Correct Answer: 3,4