A rod AB of length 15,cm rests in between two coordinate axes in such a way that end point A lies on

English

Gujarati

Hindi

10-2. Parabola, Ellipse, Hyperbola

hard

A rod $AB$ of length $15\,cm$ rests in between two coordinate axes in such a way that the end point A lies on $x-$ axis and end point $B$ lies on $y-$ axis. A point $P(x,\, y)$ is taken on the rod in such a way that $AP =6\, cm .$ Show that the locus of $P$ is an ellipse.

Option A

Option B

Option C

Option D

Solution

Let $AB$ be the rod making an angle $\theta $ with $OX$ as shown in Fig. and $P (x, \,y)$ the point on it such that $AP =6\, cm$

since $AB =15\,cm ,$ we have

$P B=9\, cm$

From $P$ draw $PQ$ and $PR$ perpendiculars on $y-$ axis and $x-$ axis, respectively.

From $\Delta PBQ$ , $\cos \theta=\frac{x}{9}$

From $\Delta PRA$, $\sin \theta=\frac{y}{6}$

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

$\left(\frac{x}{9}\right)^{2}+\left(\frac{y}{6}\right)^{2}=1$

or $\frac{x^{2}}{81}+\frac{y^{2}}{36}=1$

Thus the locus of $P$ is an cllipse.

Standard 11

Mathematics

Share

Similar Questions

Let $E_1: \frac{x^2}{9}+\frac{y^2}{4}=1$ be an ellipse. Ellipses $E_i$ 's are constructed such that their centres and eccentricities are same as that of $E _1$, and the length of minor axis of $E _{ i }$ is the length of major axis of $E _{ i +1}( i \geq 1)$. If $A _{ i }$ is the area of the ellipse $E _{ i }$, then $\frac{5}{\pi}\left(\sum_{ i =1}^{\infty} A _{ i }\right)$, is equal to _____

normal

(JEE MAIN-2025)

The equation of the ellipse whose centre is $(2, -3)$, one of the foci is $(3, -3)$ and the corresponding vertex is $(4, -3)$ is

medium

In a group of $100$ persons $75$ speak English and $40$ speak Hindi. Each person speaks at least one of the two languages. If the number of persons, who speak only English is $\alpha$ and the number of persons who speak only Hindi is $\beta$, then the eccentricity of the ellipse $25\left(\beta^2 x^2+\alpha^2 y^2\right)=\alpha^2 \beta^2$ is $…….$

medium

(JEE MAIN-2023)

Let $E_1$ and $E_2$ be two ellipses whose centers are at the origin. The major axes of $E_1$ and $E_2$ lie along the $x$-axis and the $y$-axis, respectively. Let $S$ be the circle $x^2+(y-1)^2=2$. The straight line $x+y=3$ touches the curves $S, E_1$ ad $E_2$ at $P, Q$ and $R$, respectively. Suppose that $P Q=P R=\frac{2 \sqrt{2}}{3}$. If $e_1$ and $e_2$ are the eccentricities of $E_1$ and $E_2$, respectively, then the correct expression$(s)$ is(are)

$(A)$ $e_1^2+e_2^2=\frac{43}{40}$

$(B)$ $e_1 e_2=\frac{\sqrt{7}}{2 \sqrt{10}}$

$(C)$ $\left|e_1^2-e_2^2\right|=\frac{5}{8}$

$(D)$ $e_1 e_2=\frac{\sqrt{3}}{4}$

medium

(IIT-2015)

If $y = mx + c$ is tangent on the ellipse $\frac{{{x^2}}}{9} + \frac{{{y^2}}}{4} = 1$, then the value of $c$ is

easy