Question

$AB$ is a variable line sliding between the co-ordinate axes in such a way that $A$ lies on X-axis and $B$ lies on Y-axis. If $P$ is a variable point on $AB$ such that $PA=b,PB=a$ and $AB=a+b$, then the equation of locus of $P$ is

• A. $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$
• B. $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$
• C. $x^2+y^2=a^2+b^2$
• D. $x^2-y^2=a^2+b^2$

Answer 1

Answer: (A)

$\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$

Let p(h,k) be the point

In $\Delta PMB$,

$\sin \theta =\frac{MB}{PM}$ & $\cos \theta =\frac{b}{a}$

In $\Delta PNA$

$\sin \theta =\frac{k}{h},\cos \theta =\frac{AN}{b}$

$\therefore$ From both the ${\Delta }^{\prime }s$,

$\sin \theta =\frac{k}{b}$ & $\cos \theta =\frac{h}{a}$

$\because {\sin }^2\theta +{\cos }^2\theta =1$

${\left(\frac{k}{b}\right)}^2+{\left(\frac{h}{a}\right)}^2=1$

$\frac{a^2}{a^2}+\frac{r^2}{b^2}=1$

$\therefore \frac{x^2}{a^2}+\frac{y^2}{b^2}=1$