A line moves such that the portion of it intercepted between the coordinate axes is of constant length $a$, then the locus of the mid point of that line segment is

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Aicharya · Accepted Answer

Let $A=(p, 0)$ $B=(0, q)$ Let $ P=(h, k)$ be the mid-point of $\overline{A B}$. $\begin{array}{rlrl} \text {Given, } \overline{A B} & =a \ (h, k) & =\text { mid-point of } \overline{A B} \ (h, k) & =\left(\frac{p}{2}, \frac{q}{2}\right) \ \therefore \quad & p & =2 h, q=2 k \ \therefore \quad & A & =(p, 0)=(2 h, 0) \ & B & =(0, q)=(0,2 k) \end{array}$ Since, length of $A B=a$ $\begin{aligned} \sqrt{(2 h)^2+(2 k)^2} & =a \ \sqrt{4 h^2+4 k^2} & =a \ 4 h^2+4 k^2 & =a^2 \ h^2+k^2 & =\frac{a^2}{4} \end{aligned}$ $\therefore$ Required locus is $x^2+y^2=\frac{a^2}{4}$ $\therefore$ Hence, solution is (c).

A line moves such that the portion of it intercepted between the coordinate axes is of constant length \(a\)…

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