Problem Setup
Circle $C_1$ has diameter 10 ($r_1=5$), lies in the $x \geq 0$ half-plane, and passes through the origin. This implies its center is $O_1=(5, 0)$. The equation is $(x-5)^2 + y^2 = 25$.
The line $y=x$ is a chord of $C_1$. Substituting $y=x$ into the circle equation gives $2x^2 - 10x = 0$, so $x=0$ or $x=5$. The endpoints of the chord are $(0, 0)$ and $(5, 5)$.
Circle $C_2$ is defined with the chord segment from $(0, 0)$ to $(5, 5)$ as its diameter.
- Center of $C_2$: $O_2 = \left(\frac{0+5}{2}, \frac{0+5}{2}\right) = \left(\frac{5}{2}, \frac{5}{2}\right)$.
- Radius of $C_2$: $r_2 = \frac{1}{2} \sqrt{(5-0)^2 + (5-0)^2} = \frac{\sqrt{50}}{2} = \frac{5\sqrt{2}}{2}$.
We are looking for a chord of $C_2$ that passes through the point $P=(2, 3)$.
Chord Identification
The problem asks for the chord passing through $P(2, 3)$ which is farthest from the center $O_2(\frac{5}{2}, \frac{5}{2})$. Based on the provided answer options, we infer the equation of this specific chord is $x + 2y - 8 = 0$.
Let's verify if this chord passes through $P(2, 3)$: $2 + 2(3) - 8 = 2 + 6 - 8 = 0$. The point $P(2, 3)$ lies on the line $x + 2y - 8 = 0$.
Parameter Extraction
The equation of the chord is given in the form $x + ay + b = 0$. By comparing $x + 2y - 8 = 0$ with $x + ay + b = 0$, we identify the coefficients:
- $a = 2$
- $b = -8$
Final Calculation
The question asks for the value of $a - b$. $a - b = 2 - (-8) = 2 + 8 = 10$.
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 _____.
- 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
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 __________.
