Question

A line of fixed length (a + b) moves so that its ends are always on two fixed perpendicular straight lines. The locus of the point which divided this line into portions of lengths a & b.

• A. an ellipse
• B. an hyperbola
• C. a circle
• D. a straight line

Answer 1

Answer: (A)

an ellipse

R.E.F image

Let the two perpendicular

lines $x=0$ and $y=0$

Let the coordinates of point be (b,x)

By section formula

$h=\frac{ax+b\left(0\right)}{a+b}$ $k=\frac{a\left(0\right)+by}{a+b}$

By Pythagoras theorem

$(a+b{)}^2=x^2+y^2...(1)$

From above $x=\frac{(a+b)h}{a},y=\frac{(a+b)k}{b}$

$\therefore x^2+y^2=\frac{(a+b{)}^2{h}^2}{a^2},y=\frac{(a^2+b^2)k^2}{b^2}$

But $x^2+y^2=(a+b{)}^2$ from (1)

$\therefore \frac{(a+b{)}^2{h}^2}{a^2}+\frac{(a+b{)}^2{k}^2}{b^2}=(a+b{)}^2$

$\therefore \frac{h^2}{a^2}+\frac{k^2}{b^2}=1$

which is standard ellipse (as we took axis as x and y axis)