Circle Properties from Equation and Constraints
The given circle equation is $x^2 + y^2 + 2gx + 2fy + 25 = 0$.
- The center of the circle is $C(-g, -f)$.
- The radius squared is $r^2 = g^2 + f^2 - 25$.
- The center $(-g, -f)$ is in the first quadrant, meaning $-g > 0$ and $-f > 0$, so $g < 0$ and $f < 0$.
- The center lies on the line $2x - y = 4$. Substituting the center coordinates:
$2(-g) - (-f) = 4$
$-2g + f = 4 \implies f = 4 + 2g$
Radius Calculation from Inscribed Triangle Area
The area ($A$) of an equilateral triangle inscribed in a circle of radius $r$ is given by the formula:
$A = \frac{3\sqrt{3}}{4}r^2$
Given $A = 27\sqrt{3}$:
$27\sqrt{3} = \frac{3\sqrt{3}}{4}r^2$
Solving for $r^2$:
$r^2 = \frac{4 \times 27\sqrt{3}}{3\sqrt{3}} = 4 \times 9 = 36$
Thus, the radius $r = \sqrt{36} = 6$.
Determining Circle Center Coordinates (g and f)
Using $r^2 = 36$ in the radius formula:
$36 = g^2 + f^2 - 25$
$g^2 + f^2 = 61$
Substitute $f = 4 + 2g$ into this equation:
$g^2 + (4 + 2g)^2 = 61$
$g^2 + 16 + 16g + 4g^2 = 61$
$5g^2 + 16g - 45 = 0$
Solving the quadratic equation for $g$:
$g = \frac{-16 \pm \sqrt{16^2 - 4(5)(-45)}}{2(5)} = \frac{-16 \pm \sqrt{256 + 900}}{10} = \frac{-16 \pm \sqrt{1156}}{10} = \frac{-16 \pm 34}{10}$
The possible values for $g$ are $g = \frac{18}{10} = 1.8$ or $g = \frac{-50}{10} = -5$.
Since the center $(-g, -f)$ is in the first quadrant, $g$ must be negative ($g < 0$). Therefore, $g = -5$.
Now, find $f$:
$f = 4 + 2g = 4 + 2(-5) = 4 - 10 = -6$
The center of the circle is $C(-g, -f) = (5, 6)$. The radius is $r=6$.
Chord Length Calculation
We need to find the square of the length of the chord formed by the line $x = 1$.
Calculate the perpendicular distance ($d$) from the center $C(5, 6)$ to the line $x = 1$ (or $x - 1 = 0$):
$d = \frac{|1(5) - 1|}{\sqrt{1^2}} = \frac{|4|}{1} = 4$
Let $L$ be the length of the chord. The radius ($r$), distance ($d$), and half the chord length ($L/2$) form a right-angled triangle. Using the Pythagorean theorem, $r^2 = d^2 + (L/2)^2$:
$6^2 = 4^2 + (L/2)^2$
$36 = 16 + (L/2)^2$
$(L/2)^2 = 36 - 16 = 20$
The square of the length of the chord is $L^2$:
$L^2 = \left(2 \times \frac{L}{2}\right)^2 = 4 \times \left(\frac{L}{2}\right)^2 = 4 \times 20 = 80$
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 _________.
- 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 _____.
- 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 _________.
- 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 __________.
