The ratio of the distance of a point from a f | vedclass

Name: Exam Paper Generator | JEE NEET Paper Generator | Vedclass
Brand: Vedclass
Rating: 5 (35 reviews)

Question

(B) Let the point be $P(h, k)$. The fixed point (focus) is $S(-2, 0)$ and the directrix is $x = 9/2$,which can be written as $2x - 9 = 0$.Given the ra

Vedclass · Accepted Answer

Ellipse

Vedclass · Answer

(B) Let the point be $P(h, k)$. The fixed point (focus) is $S(-2, 0)$ and the directrix is $x = 9/2$,which can be written as $2x - 9 = 0$.Given the ratio of distances $PS : PM = 2 : 3$,where $PM$ is the perpendicular distance from $P$ to the line $2x - 9 = 0$.So,$PS = \frac{2}{3} PM$.Squaring both sides,$(PS)^2 = \frac{4}{9} (PM)^2$.$(h + 2)^2 + k^2 = \frac{4}{9} \left( \frac{2h - 9}{2} \right)^2$.$(h + 2)^2 + k^2 = \frac{4}{9} \cdot \frac{(2h - 9)^2}{4}$.$9[(h + 2)^2 + k^2] = (2h - 9)^2$.$9[h^2 + 4h + 4 + k^2] = 4h^2 - 36h + 81$.$9h^2 + 36h + 36 + 9k^2 = 4h^2 - 36h + 81$.$5h^2 + 9k^2 = 45$.Dividing by $45$,we get $\frac{h^2}{9} + \frac{k^2}{5} = 1$.The locus of the point $P(h, k)$ is $\frac{x^2}{9} + \frac{y^2}{5} = 1$,which represents an ellipse since the eccentricity $e = 2/3 < 1$.

The ratio of the distance of a point from a fixed point and a line $x = 9/2$ is always $2 : 3$. Then the locus of the point is

Similar Questions

If $\alpha, \beta$ are the eccentric angles of the extremities of a focal chord (other than the major axis) of the ellipse $x^2+4y^2=4$,then $\sqrt{3} \cos \frac{\alpha+\beta}{2} =$

The locus of a point such that the sum of its distances from the points $(0, 2)$ and $(0, -2)$ is $6$ is:

The number of values of $c$ such that the straight line $y = 4x + c$ touches the curve $\frac{x^2}{4} + y^2 = 1$ is

Find the coordinates of the foci,the vertices,the length of the major axis,the minor axis,the eccentricity,and the length of the latus rectum of the ellipse $\frac{x^{2}}{36}+\frac{y^{2}}{16}=1$.

Tangents are drawn to the ellipse $\frac{x^2}{25}+\frac{y^2}{16}=1$ at all the four ends of its latus rectum. The area (in sq units) of the quadrilateral formed by these tangents is

Vedclass Test Series

Exam Paper Generator

Online Exam Module