S = (-1,1) is the focus, \( 2x - 3y + 1 = 0 \) is the directrix corresponding to S and \( \frac{1}{2} \) is the eccentricity of an ellipse. If \( (a,b) \) is the centre of the ellipse, then \( 3a + 2b \) is:
\( 0 \)
Step 1: Using the Formula for the Centre of the Ellipse
The formula for the centre of an ellipse given a focus \( S(h,k) \), directrix \( Ax + By + C = 0 \), and eccentricity \( e \) is: \[ \frac{|Ax + By + C|}{\sqrt{A^2 + B^2}} = e \times \text{distance of the centre from the focus}. \] Here, we have: - Focus \( S(-1,1) \). - Directrix: \( 2x - 3y + 1 = 0 \). - Eccentricity: \( e = \frac{1}{2} \). Using the formula for the centre of the ellipse: \[ (a,b) = \left(\frac{-1 + \lambda 2}{1 + \lambda^2}, \frac{1 + \lambda (-3)}{1 + \lambda^2} \right). \] Substituting \( e = \frac{1}{2} \), solving for \( a, b \), and substituting in \( 3a + 2b \): \[ 3a + 2b = -1. \]
Step 2: Conclusion
Thus, the final answer is: \[ \boxed{-1}. \]
A rectangle is formed by the lines \[ x = 4, \quad x = -2, \quad y = 5, \quad y = -2 \] and a circle is drawn through the vertices of this rectangle. The pole of the line \[ y + 2 = 0 \] with respect to this circle is:
The equation of a circle which passes through the points of intersection of the circles \[ 2x^2 + 2y^2 - 2x + 6y - 3 = 0, \quad x^2 + y^2 + 4x + 2y + 1 = 0 \] and whose centre lies on the common chord of these circles is:
The following graph is obtained for the adsorption of a gas on the surface of a catalyst. The values of k and n are respectively
Observe the following reactions (not balanced):
Cl2 + NaOH → NaCl + X + H2O
Cl2 + NaOH → NaCl + Y + H2O
Which of the following is not correct?
(1) XeO2 is a colorless explosive gas
(2) SO2 is highly soluble in water
(3) Noble gases have very low boiling points
(4) The boiling point of sulphur is more than that of oxygen