Question:
The equation of the ellipse whose axes are parallel to the coordinate axes having its centre at the point \( (2, -3) \) and focus at \( (3, -3) \) and one vertex at \( (4, -3) \) is
The equation of the ellipse whose axes are parallel to the coordinate axes having its centre at the point \( (2, -3) \) and focus at \( (3, -3) \) and one vertex at \( (4, -3) \) is
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For ellipses with axes parallel to the coordinate axes, use the standard form and the relationship \( c^2 = a^2 - b^2 \) to find the equation.
Updated On: Apr 22, 2026
- \( \frac{(x - 2)^2}{4} + \frac{(y + 3)^2}{3} = 1 \)
- \( \frac{(x + 2)^2}{3} + \frac{(y + 3)^2}{4} = 1 \)
- \( \frac{(x - 2)^2}{4} + \frac{(y + 3)^2}{3} = 1 \)
- None of the above
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The Correct Option is A
Solution and Explanation
Step 1: Understand the geometry of the ellipse.
The standard form of an ellipse with its center at \( (h, k) \) and axes parallel to the coordinate axes is: \[ \frac{(x - h)^2}{a^2} + \frac{(y - k)^2}{b^2} = 1 \] where \( a \) is the length of the semi-major axis and \( b \) is the length of the semi-minor axis. The foci of the ellipse are located at \( (h \pm c, k) \) if the major axis is horizontal or at \( (h, k \pm c) \) if the major axis is vertical.
Step 2: Analyze the given points.
We are given:
- The center of the ellipse at \( (2, -3) \),
- One vertex at \( (4, -3) \),
- One focus at \( (3, -3) \). The distance between the center and the vertex is the length of the semi-major axis \( a \), and the distance between the center and the focus is the focal distance \( c \).
Step 3: Calculate \( a \) and \( c \).
The distance between the center \( (2, -3) \) and the vertex \( (4, -3) \) is 2 units, so: \[ a = 2 \] The distance between the center \( (2, -3) \) and the focus \( (3, -3) \) is 1 unit, so: \[ c = 1 \]
Step 4: Use the relationship \( c^2 = a^2 - b^2 \).
For an ellipse, the relationship between \( a \), \( b \), and \( c \) is given by: \[ c^2 = a^2 - b^2 \] Substitute \( a = 2 \) and \( c = 1 \) into this equation: \[ 1^2 = 2^2 - b^2 \quad \implies \quad 1 = 4 - b^2 \quad \implies \quad b^2 = 3 \]
Step 5: Write the equation of the ellipse.
Thus, the equation of the ellipse is: \[ \frac{(x - 2)^2}{2^2} + \frac{(y + 3)^2}{\sqrt{3}^2} = 1 \] Simplifying: \[ \frac{(x - 2)^2}{4} + \frac{(y + 3)^2}{3} = 1 \]
Step 6: Conclusion.
Thus, the equation of the ellipse is \( \frac{(x - 2)^2}{4} + \frac{(y + 3)^2}{3} = 1 \), corresponding to option (A).
The standard form of an ellipse with its center at \( (h, k) \) and axes parallel to the coordinate axes is: \[ \frac{(x - h)^2}{a^2} + \frac{(y - k)^2}{b^2} = 1 \] where \( a \) is the length of the semi-major axis and \( b \) is the length of the semi-minor axis. The foci of the ellipse are located at \( (h \pm c, k) \) if the major axis is horizontal or at \( (h, k \pm c) \) if the major axis is vertical.
Step 2: Analyze the given points.
We are given:
- The center of the ellipse at \( (2, -3) \),
- One vertex at \( (4, -3) \),
- One focus at \( (3, -3) \). The distance between the center and the vertex is the length of the semi-major axis \( a \), and the distance between the center and the focus is the focal distance \( c \).
Step 3: Calculate \( a \) and \( c \).
The distance between the center \( (2, -3) \) and the vertex \( (4, -3) \) is 2 units, so: \[ a = 2 \] The distance between the center \( (2, -3) \) and the focus \( (3, -3) \) is 1 unit, so: \[ c = 1 \]
Step 4: Use the relationship \( c^2 = a^2 - b^2 \).
For an ellipse, the relationship between \( a \), \( b \), and \( c \) is given by: \[ c^2 = a^2 - b^2 \] Substitute \( a = 2 \) and \( c = 1 \) into this equation: \[ 1^2 = 2^2 - b^2 \quad \implies \quad 1 = 4 - b^2 \quad \implies \quad b^2 = 3 \]
Step 5: Write the equation of the ellipse.
Thus, the equation of the ellipse is: \[ \frac{(x - 2)^2}{2^2} + \frac{(y + 3)^2}{\sqrt{3}^2} = 1 \] Simplifying: \[ \frac{(x - 2)^2}{4} + \frac{(y + 3)^2}{3} = 1 \]
Step 6: Conclusion.
Thus, the equation of the ellipse is \( \frac{(x - 2)^2}{4} + \frac{(y + 3)^2}{3} = 1 \), corresponding to option (A).
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