MCQ
A point ratio of whose distance from a fixed point and line $x = 9/2$ is always $2 : 3$. Then locus of the point will be
- AHyperbola
- ✓Ellipse
- CParabola
- DCircle
✓
Answer
Correct option: B.
Ellipse
b
(b) In question, $PS = \frac{2}{3}PM$ (Given)
(b) In question, $PS = \frac{2}{3}PM$ (Given)
Focus $S( - 2,\,0)$,
Equation of directrix $2x - 9 = 0$
${(PS)^2} = \frac{4}{9}{(PM)^2}$
==> ${(h + 2)^2} + {(k)^2} = \frac{4}{9}{\left( {\frac{{2h - 9}}{2}} \right)^2}$
==> $9[{(h + 2)^2} + {(k)^2}] = \frac{{4{{(2h - 9)}^2}}}{4}$
==> $9{h^2} + 9{k^2} + 36h + 36 = 4{h^2} + 81 + 36h$
==> $\frac{{5{h^2}}}{{45}} + \frac{{9{k^2}}}{{45}} = 1$
==> $\frac{{{h^2}}}{9} + \frac{{{k^2}}}{5} = 1$
Locus of point $P(h, k)$ is $\frac{{{x^2}}}{9} + \frac{{{y^2}}}{5} = 1$, which is an ellipse
Need a full question paper?
Generate a complete, print-ready paper with questions like this in minutes — across 16+ boards, with answer keys.
Explore more
Similar questions
The value of $(1+\text{i})^4+(1-\text{i})^4$ is:
→If the distances from the origin to the centres of the three circles ${x^2} + {y^2} - 2{\lambda _i}\,x = {c^2},(i = 1,\,2,\,3)$ are in $G. P.$, then the lengths of the tangents drawn to them from any point on the circle ${x^2} + {y^2} = {c^2}$ are in
→Number of common tangents of the ellipse $\frac{{{{\left( {x - 2} \right)}^2}}}{9} + \frac{{{{\left( {y + 2} \right)}^2}}}{4} = 1$ and the circle $x^2 + y^2 -4x + 2y + 4 = 0$ is
→Let $a_1=8, a_2, a_3, \ldots a_n$ be an $A.P.$ If the sum of its first four terms is $50$ and the sum of its last four terms is $170$ , then the product of its middle two terms is
→Coefficient of ${x^2}$ in the expansion of ${\left( {x - \frac{1}{{2x}}} \right)^8}$ is
→8, 24, 48, 80, 120, .....:
Let $a_1, a_2, \ldots \ldots, a_n$ be in A.P. If $a_5=2 a_3$ and $a_{11}=18$, then $12\left(\frac{1}{\sqrt{a_{10}}+\sqrt{a_{11}}}+\frac{1}{\sqrt{a_{11}}+\sqrt{a_{12}}}+\ldots . \cdot \frac{1}{\sqrt{a_{17}}+\sqrt{a_{18}}}\right)$ is equal to $..........$.
→If $x = {\log _a}(bc),y = {\log _b}(ca),z = {\log _c}(ab),$then which of the following is equal to $1$
→A sequence is called ___________________ if $a_{n+1}=a_n \times r$.
→If three positive numbers $a, b$ and $c$ are in $A.P.$ such that $abc\, = 8$, then the minimum possible value of $b$ is
→