Let P (x1, y1) and Q (x2, y2) be the points of trisection of the line segment joining the given points i.e., AP = PQ = QB
Therefore, point P divides AB internally in the ratio 1:2.
\(x_1=\frac{1\times(-2)+2\times4}{1+2}\), \(y_1=\frac{1\times (-3)+2\times(-1)}{1+2}\)
\(x_1=\frac{-2+8}{3}=\frac{6}{3}=2\), \(y_1=\frac{-3-2}{3}=\frac{-5}{3}\)
Point Q divides AB internally in the ratio of 2:1.
\(x_2=\frac{2\times(-2)+1\times4}{2+1}\) , \(y_2=\frac{2\times(-3)+1\times(-1)}{2+1}\)
\(x_2=\frac{-4+4}{3}=0\) , \(y_2=\frac{-6-1}{3}=\frac{-7}{3}\)
Q (x2, y2) = \((0,-\frac{7}{3})\)
Let \( F \) and \( F' \) be the foci of the ellipse \( \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \) (where \( b<2 \)), and let \( B \) be one end of the minor axis. If the area of the triangle \( FBF' \) is \( \sqrt{3} \) sq. units, then the eccentricity of the ellipse is:
A common tangent to the circle \( x^2 + y^2 = 9 \) and the parabola \( y^2 = 8x \) is
If the equation of the circle passing through the points of intersection of the circles \[ x^2 - 2x + y^2 - 4y - 4 = 0, \quad x^2 + y^2 + 4y - 4 = 0 \] and the point \( (3,3) \) is given by \[ x^2 + y^2 + \alpha x + \beta y + \gamma = 0, \] then \( 3(\alpha + \beta + \gamma) \) is: