Finding Ellipse Equation from Circle Transformation
We are given that the point $(h, k)$ lies on the circle $C$ defined by the equation $x^2 + y^2 = 4$. This means $h^2 + k^2 = 4$.
A second point $(x', y') = (2h + 1, 3k + 2)$ lies on an ellipse.
We need to find the equation of the ellipse by expressing $h$ and $k$ in terms of $x'$ and $y'$ and substituting them into the circle equation.
Deriving Ellipse Equation
-
Express $h$ and $k$ from the coordinates $(x', y')$:
From $x' = 2h + 1$, we get $h = \frac{x' - 1}{2}$.
From $y' = 3k + 2$, we get $k = \frac{y' - 2}{3}$.
-
Substitute these expressions for $h$ and $k$ into the circle equation $h^2 + k^2 = 4$:
$ \left(\frac{x' - 1}{2}\right)^2 + \left(\frac{y' - 2}{3}\right)^2 = 4 $
-
Simplify the equation:
$ \frac{(x' - 1)^2}{4} + \frac{(y' - 2)^2}{9} = 4 $
Divide the entire equation by 4 to get the standard form:
$ \frac{(x' - 1)^2}{16} + \frac{(y' - 2)^2}{36} = 1 $
This is the equation of the ellipse.
Calculating Ellipse Eccentricity
The equation of the ellipse is $\frac{(x - 1)^2}{16} + \frac{(y - 2)^2}{36} = 1$.
Comparing this to the standard form $\frac{(x-h_0)^2}{a^2} + \frac{(y-k_0)^2}{b^2} = 1$, we identify:
- Semi-minor axis squared: $a^2 = 16 \implies a = 4$.
- Semi-major axis squared: $b^2 = 36 \implies b = 6$.
Since $b^2 > a^2$, the major axis is vertical.
The formula for the square of the eccentricity ($e^2$) for such an ellipse is:
$ e^2 = 1 - \frac{a^2}{b^2} $
Substitute the values of $a^2$ and $b^2$:
$ e^2 = 1 - \frac{16}{36} = 1 - \frac{4}{9} = \frac{5}{9} $
Final Calculation
The question asks for the value of $\frac{5}{e^2}$.
Using the calculated value $e^2 = \frac{5}{9}$:
$ \frac{5}{e^2} = \frac{5}{(5/9)} = 5 \times \frac{9}{5} = 9 $
The value is exactly 9.
Similar Questions
- Let a line L passing through the point $P(1, 1, 1)$ be perpendicular to the lines $\frac{x-4}{4} = \frac{y-1}{1} = \frac{z-1}{1}$ and $\frac{x-17}{1} = \frac{y-71}{1} = \frac{z}{0}$. Let the line L intersect the yz-plane at the point Q. Another line parallel to L and passing through the point $S(1, 0, -1)$ intersects the yz-plane at the point R. Then the square of the area of the parallelogram PQRS is equal to ______.
- Let the angles made with the positive $x$-axis by two straight lines drawn from the point $P(2, 3)$ and meeting the line $x + y = 6$ at a distance $\sqrt{\frac{2}{3}}$ from the point $P$ be $\theta_1$ and $\theta_2$. Then the value of $(\theta_1 + \theta_2)$ is:
- The sum of all values of $\alpha$, for which the shortest distance between the lines $\frac{x + 1}{\alpha} = \frac{y - 2}{-1} = \frac{z - 4}{-\alpha}$ and $\frac{x}{\alpha} = \frac{y - 1}{2} = \frac{z - 1}{2\alpha}$ is $\sqrt{2}$, is
- Let the length of the latus rectum of an ellipse $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1, (a > b)$, be 30. If its eccentricity is the maximum value of the function $f(t) = -\frac{3}{4} + 2t - t^2$, then $(a^2 + b^2)$ is equal to
- Let $\vec{a} = 2\hat{i} - \hat{j} - \hat{k}$, $\vec{b} = \hat{i} + 3\hat{j} - \hat{k}$ and $\vec{c} = 2\hat{i} + \hat{j} + 3\hat{k}$. Let $\vec{v}$ be the vector in the plane of the vectors $\vec{a}$ and $\vec{b}$, such that the length of its projection on the vector $\vec{c}$ is $\frac{1}{\sqrt{14}}$. Then $|\vec{v}|$ is equal to
- Let $\vec{a} = 2\hat{i} - 5\hat{j} + 5\hat{k}$ and $\vec{b} = \hat{i} - \hat{j} + 3\hat{k}$. If $\vec{c}$ is a vector such that $2(\vec{a} \times \vec{c}) + 3(\vec{b} \times \vec{c}) = \vec{0}$ and $(\vec{a} - \vec{b}) \cdot \vec{c} = -97$, then $|\vec{c} \times \hat{k}|^2$ is equal to
- Let the image of parabola $x^2 = 4y$, in the line $x - y = 1$ be $(y + a)^2 = b(x - c)$, $a, b, c \in \mathbb{N}$. Then $a + b + c$ is equal to
- If $P$ is a point on the circle $x^2 + y^2 = 4$, $Q$ is a point on the straight line $5x + y + 2 = 0$ and $x - y + 1 = 0$ is the perpendicular bisector of $PQ$, then $13$ times the sum of abscissa of all such points $P$ is __________.
- If the distance of the point $\text{P}(43, \alpha, \beta)$, $\beta < 0$, from the line $\overrightarrow{\text{r}} = 4\hat{i} - \hat{k} + \mu(2\hat{i} + 3\hat{k})$, $\mu \in \mathbf{R}$ along a line with direction ratios $3, -1, 0$ is $13\sqrt{10}$, then $\alpha^{2} + \beta^{2}$ is equal to _________.
- From the point $(-1, -1)$, two rays are sent making angles of $45^{\circ}$ with the line $x+y=0$. These rays get reflected from the mirror $x+2y=1$. If the equations of the reflected rays are $ax+by=9$ and $cx+dy=7$, $a, b, c, d \in \mathbf{Z}$, then the value of $\text{ad}+\text{bc}$ is ___________.
Important Questions from Coordinate Geometry
- Let a line L passing through the point $P(1, 1, 1)$ be perpendicular to the lines $\frac{x-4}{4} = \frac{y-1}{1} = \frac{z-1}{1}$ and $\frac{x-17}{1} = \frac{y-71}{1} = \frac{z}{0}$. Let the line L intersect the yz-plane at the point Q. Another line parallel to L and passing through the point $S(1, 0, -1)$ intersects the yz-plane at the point R. Then the square of the area of the parallelogram PQRS is equal to ______.
- Let the angles made with the positive $x$-axis by two straight lines drawn from the point $P(2, 3)$ and meeting the line $x + y = 6$ at a distance $\sqrt{\frac{2}{3}}$ from the point $P$ be $\theta_1$ and $\theta_2$. Then the value of $(\theta_1 + \theta_2)$ is:
- The sum of all values of $\alpha$, for which the shortest distance between the lines $\frac{x + 1}{\alpha} = \frac{y - 2}{-1} = \frac{z - 4}{-\alpha}$ and $\frac{x}{\alpha} = \frac{y - 1}{2} = \frac{z - 1}{2\alpha}$ is $\sqrt{2}$, is
- Let the length of the latus rectum of an ellipse $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1, (a > b)$, be 30. If its eccentricity is the maximum value of the function $f(t) = -\frac{3}{4} + 2t - t^2$, then $(a^2 + b^2)$ is equal to
- Let $\vec{a} = 2\hat{i} - \hat{j} - \hat{k}$, $\vec{b} = \hat{i} + 3\hat{j} - \hat{k}$ and $\vec{c} = 2\hat{i} + \hat{j} + 3\hat{k}$. Let $\vec{v}$ be the vector in the plane of the vectors $\vec{a}$ and $\vec{b}$, such that the length of its projection on the vector $\vec{c}$ is $\frac{1}{\sqrt{14}}$. Then $|\vec{v}|$ is equal to
