Understanding the Locus of Point B
We are given a fixed point A = $(3, 0)$ and a fixed circle $x^2 + y^2 = 36$ with center O = $(0, 0)$ and radius $R = 6$. Let the point B have coordinates $(x, y)$. The circle with diameter AB touches the fixed circle internally.
Variable Circle Properties
The center M of the variable circle (with diameter AB) is the midpoint of AB:
M = $\left(\frac{x+3}{2}, \frac{0+y}{2}\right) = \left(\frac{x+3}{2}, \frac{y}{2}\right)$
The radius $r$ of this variable circle is half the length of the diameter AB:
$r = \frac{1}{2} \text{length}(AB) = \frac{1}{2} \sqrt{(x-3)^2 + (y-0)^2} = \frac{1}{2} \sqrt{(x-3)^2 + y^2}$
Internal Tangency Condition
For the variable circle to touch the fixed circle internally, the distance between their centers (OM) must equal the difference between their radii ($R - r$):
Distance OM = $\sqrt{\left(\frac{x+3}{2} - 0\right)^2 + \left(\frac{y}{2} - 0\right)^2} = R - r$
Substituting the expressions for OM, R, and r:
$\sqrt{\frac{(x+3)^2}{4} + \frac{y^2}{4}} = 6 - \frac{1}{2} \sqrt{(x-3)^2 + y^2}$
Deriving the Locus Equation
Simplify the equation:
$\frac{1}{2} \sqrt{(x+3)^2 + y^2} = 6 - \frac{1}{2} \sqrt{(x-3)^2 + y^2}$
Multiply both sides by 2:
$\sqrt{(x+3)^2 + y^2} = 12 - \sqrt{(x-3)^2 + y^2}$
Rearrange the terms:
$\sqrt{(x+3)^2 + y^2} + \sqrt{(x-3)^2 + y^2} = 12$
Identifying the Locus Curve C
The equation $\sqrt{(x+3)^2 + y^2} + \sqrt{(x-3)^2 + y^2} = 12$ is the standard definition of an ellipse.
It represents the set of all points $(x, y)$ such that the sum of the distances from $(x, y)$ to two fixed points (the foci) is constant.
Here, the foci are $F_1 = (3, 0)$ and $F_2 = (-3, 0)$.
Calculating Ellipse Parameters
From the standard form of an ellipse with foci at $(\pm c, 0)$ and the sum of distances $2a$:
- The distance from the center (0,0) to each focus is $c = 3$.
- The constant sum of distances is $2a = 12$, which gives $a = 6$.
Finding Eccentricity (e)
The eccentricity $e$ of an ellipse is defined as $e = \frac{c}{a}$.
Using the values calculated:
$e = \frac{3}{6} = \frac{1}{2}$
Final Calculation
The question asks for the value of $72e^2$.
$72e^2 = 72 \times \left(\frac{1}{2}\right)^2$
$72e^2 = 72 \times \frac{1}{4}$
$72e^2 = 18$
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 $(h, k)$ lie on the circle $C : x^2 + y^2 = 4$ and the point $(2h + 1, 3k + 2)$ lie on an ellipse with eccentricity $e$. Then the value of $\frac{5}{e^2}$ 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 _________.
- 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 ______.
- Let $(h, k)$ lie on the circle $C : x^2 + y^2 = 4$ and the point $(2h + 1, 3k + 2)$ lie on an ellipse with eccentricity $e$. Then the value of $\frac{5}{e^2}$ 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 _________.
- 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 _____.
