Question:
If A and B are the points of intersection of the circles \( x^2 + y^2 - 4x + 6y - 3 = 0 \) and \( x^2 + y^2 + 2x - 2y - 2 = 0 \), then the distance between A and B is:
If A and B are the points of intersection of the circles \( x^2 + y^2 - 4x + 6y - 3 = 0 \) and \( x^2 + y^2 + 2x - 2y - 2 = 0 \), then the distance between A and B is:
Show Hint
To find the distance between the intersection points of two circles, use the geometry of the system of equations and apply the distance formula.
Updated On: May 13, 2025
- \( \frac{13}{10} \)
- \( \frac{\sqrt{41}}{3} \)
- \( \frac{\sqrt{231}}{5} \)
- \( \frac{26}{5} \)
Hide Solution
Verified By Collegedunia
The Correct Option is C
Solution and Explanation
Step 1: Equations of the circles.
The equations of the two circles are: \[ x^2 + y^2 - 4x + 6y - 3 = 0 \] and \[ x^2 + y^2 + 2x - 2y - 2 = 0 \]
Step 2: Subtract the two equations.
Subtract the second equation from the first: \[ (x^2 + y^2 - 4x + 6y - 3) - (x^2 + y^2 + 2x - 2y - 2) = 0 \] Simplifying: \[ -6x + 8y - 1 = 0 \quad \Rightarrow \quad 3x - 4y = \frac{1}{2} \]
Step 3: Find the distance between the points.
Using the distance formula between the points of intersection of two circles, we find the distance between A and B to be: \[ \text{Distance} = \frac{\sqrt{231}}{5} \]
The equations of the two circles are: \[ x^2 + y^2 - 4x + 6y - 3 = 0 \] and \[ x^2 + y^2 + 2x - 2y - 2 = 0 \]
Step 2: Subtract the two equations.
Subtract the second equation from the first: \[ (x^2 + y^2 - 4x + 6y - 3) - (x^2 + y^2 + 2x - 2y - 2) = 0 \] Simplifying: \[ -6x + 8y - 1 = 0 \quad \Rightarrow \quad 3x - 4y = \frac{1}{2} \]
Step 3: Find the distance between the points.
Using the distance formula between the points of intersection of two circles, we find the distance between A and B to be: \[ \text{Distance} = \frac{\sqrt{231}}{5} \]
Was this answer helpful?
0
0
Top Questions on Geometry
- If \( P(10, 2\sqrt{15}) \) lies on the hyperbola \( \frac{x^2}{a^2} - \frac{y^2}{b^2} = 1 \) and the length of the latus rectum is 8, then the square of the area of \( \Delta PS_1S_2 \) is [where \( S_1 \) and \( S_2 \) are the foci of the hyperbola].
- A rectangle is formed by the lines \( x = 0,\ y = 0,\ x = 3,\ y = 4 \). A line perpendicular to \( 3x + 4y + 6 = 0 \) divides the rectangle into two equal parts. Then the distance of the line from the point \( \left(-1,\frac{3}{2}\right) \) is:
- If \[ \lim_{x \to 0} \frac{e^{a(x-1)} + 2\cos(bx) + e^{-x}(c - 1)}{x \cos x - \ln(1 + x)} = 2, \] Then the value of \( a^2 + b^2 + c^2 \) is:
- If \( \cos(\alpha + \beta) = -\frac{1}{10} \) and \( \sin(\alpha - \beta) = \frac{3}{8} \), where \[ 0<\alpha<\frac{\pi}{3} \quad \text{and} \quad 0<\beta<\frac{\pi}{4}, \] and \[ \tan(2\alpha) = \frac{3\left(1 - \sqrt{55}\right)}{\sqrt{11} \left(s + \sqrt{5}\right)}, \] and \( r, s \in \mathbb{N} \), then \( r^2 + s \) is:
- If \( O \) is the vertex of the parabola \( x^2 = 4y \), \( Q \) is a point on the parabola. If \( C \) is the locus of a point which divides \( OQ \) in the ratio \( 2:3 \), then the equation of the chord of \( C \) which is bisected at the point \( (1,2) \) is:
View More Questions
Questions Asked in AP EAPCET exam
- The ratio of the wavelengths of the first Lyman line and the second Balmer line of hydrogen atom is
- AP EAPCET - 2025
- Nuclear physics
- The incorrect statement about amylose is:
- AP EAPCET - 2025
- Biomolecules
- Which one of the following acts as an autocatalyst during titration of \( KMnO_4 \) and oxalic acid in presence of dilute \( H_2SO_4 \)?
- AP EAPCET - 2025
- Catalysis
- Consider the following statements: Statement-I: Primary structure of protein represents its constitution. Statement-II: \( \alpha \)-Helix and \( \beta \)-pleated sheet structure of protein represent tertiary structure of it. The correct answer is:
- AP EAPCET - 2025
- Biomolecules
How many moles of \(\mathrm{CO_2}\) are produced when 4 moles of \(\mathrm{C_2H_6}\) (ethane) combust completely?
Given reaction: \[ 2 \mathrm{C_2H_6} + 7 \mathrm{O_2} \rightarrow 4 \mathrm{CO_2} + 6 \mathrm{H_2O} \]- AP EAPCET - 2025
- Stoichiometry and Stoichiometric Calculations
View More Questions