Andhra Pradesh Board Solutions for Exercise 1: Exercise 4(a)

Find the equation of the ellipse in the form $\frac{(x-h{)}^2}{a^2}+\frac{(y-k{)}^2}{b^2}=1$, given center $(2,-1)$, one end of the major axis $(2,-5) \mathrm{and} e=\frac{1}{3}$.

Find the equation of the ellipse in the form $\frac{(x-h{)}^2}{a^2}+\frac{(y-k{)}^2}{b^2}=1$, given center $(4,-1)$, one end of the major axis $(-1,-1)$ and passes through $\left(8,0\right)$

Find the equation of the ellipse in the form $\frac{(x-h{)}^2}{a^2}+\frac{(y-k{)}^2}{b^2}=1$, given centre $(0,-3)$, $e=\frac{2}{3}$, semi minor axis $5$.

Find the equation of the ellipse in the form $\frac{(x-h{)}^2}{a^2}+\frac{(y-k{)}^2}{b^2}=1$, given center $\left(2,-1\right)$, $e=\frac{1}{2}$ and length of latus rectum $4$

Find the radius of the circle passing through the foci of an ellipse $9x^2+16y^2=144$ and having least radius $\sqrt{k}$

A man running on a race course notices that the sum of the distances of the two flag posts from him is always $10m$, and the distance between the flag posts is $8m$. Find the equation of the race course traced by the man.

A line of fixed length $\left(a+b\right)$ moves so that its ends are always on two fixed perpendicular straight lines. Prove that a marked point on the line, which divides this line into portions of length $\text{'}a\text{'}$ and $\text{'}b\text{'}$ describes an ellipse and also find the eccentricity of the ellipse when $a=8$, $b=12$.

Prove that the equation of the chord joining the points $\alpha$ and $\beta$ on the ellipse $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$ is $\frac{x}{a}\cos \left(\frac{\alpha +\beta }{2}\right)+\frac{y}{b}\sin \left(\frac{\alpha +\beta }{2}\right)=\cos \left(\frac{\alpha -\beta }{2}\right)$