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
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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 $k = \frac{y' - 2}{3}$.
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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 $
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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 ______.
- 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 _________.
- Let A be the point $(3, 0)$ and circles with variable diameter AB touch the circle $x^2 + y^2 = 36$ internally. Let the curve C be the locus of the point B. If the eccentricity of C is e, then $72e^2$ is equal to __________.
- Consider the parabola $\text{P} : y^2 = 4kx$ and the ellipse $\text{E} : \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$. Let the line segment joining the points of intersection of P and E, be their latus rectums. If the eccentricity of E is $e$, then $e^2 + 2\sqrt{2}$ is equal to _____.
- Let the centre of the circle $x^2 + y^2 + 2gx + 2fy + 25 = 0$ be in the first quadrant and lie on the line $2x - y = 4$. Let the area of an equilateral triangle inscribed in the circle be $27\sqrt{3}$. Then the square of the length of the chord of the circle on the line $x = 1$ is_________.
- If $\vec{a} = \hat{i} + \hat{j} + \hat{k}$, $\vec{b} = \hat{j} - \hat{k}$ and $\vec{c}$ be three vectors such that $\vec{a} \times \vec{c} = \vec{b}$ and $\vec{a} \cdot \vec{c} = 3$, then $\vec{c} \cdot (\vec{a} - 2\vec{b})$ is equal to _________.
- For some $\theta \in \left(0, \frac{\pi}{2}\right)$, let the eccentricity and the length of the latus rectum of the hyperbola $x^2 - y^2\sec^2\theta = 8$ be $e_1$ and $l_1$, respectively, and let the eccentricity and the length of the latus rectum of the ellipse $x^2\sec^2\theta + y^2 = 6$ be $e_2$ and $l_2$, respectively. If $e_1^2 = e_2^2(\sec^2\theta+1)$, then $\left(\frac{l_1 l_2}{e_1 e_2}\right)\tan^2\theta$ is equal to ________
- If foci of the hyperbola $\frac{x^2}{a^2} - \frac{y^2}{12} = 1$ and the ellipse $\frac{x^2}{25} + \frac{y^2}{9} = 1$ coincide, then the eccentricity of the hyperbola, with foci $(0, \pm \sqrt{10})$ passing through the point $(a, a + 1)$, is
- Let the curves $y^2 = kx$ and $xy = -1$ have a common tangent, whose slope is $\frac{1}{2}$. Then $k$ can NOT lie in the interval
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 ______.
- 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 _________.
- Let A be the point $(3, 0)$ and circles with variable diameter AB touch the circle $x^2 + y^2 = 36$ internally. Let the curve C be the locus of the point B. If the eccentricity of C is e, then $72e^2$ is equal to __________.
- Consider the parabola $\text{P} : y^2 = 4kx$ and the ellipse $\text{E} : \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$. Let the line segment joining the points of intersection of P and E, be their latus rectums. If the eccentricity of E is $e$, then $e^2 + 2\sqrt{2}$ is equal to _____.
