Let O be the origin and OP and OQ be the tangents to the circle \( x^2 + y^2 - 6x + 4y + 8 = 0 \) at the points P and Q on it. If the circumcircle of the triangle OPQ passes through the point \( \left( \alpha, \frac{1}{2} \right) \), then a value of \( \alpha \) is.
Let O be the origin and OP and OQ be the tangents to the circle \( x^2 + y^2 - 6x + 4y + 8 = 0 \) at the points P and Q on it. If the circumcircle of the triangle OPQ passes through the point \( \left( \alpha, \frac{1}{2} \right) \), then a value of \( \alpha \) is.
\(\frac{5}{2}\)
\(-\frac{1}{2}\)
1
\(\frac{3}{2}\)
The Correct Option is A
Solution and Explanation
Given:
Circumcircle of \( \Delta OPQ \) is described by the equation:
\[ (x - 0)(x - 3) + (y - 0)(y + 2) = 0 \] or equivalently:
\[ x^2 + y^2 - 3x + 2y = 0 \]
The circumcircle passes through \( \left( \alpha, \frac{1}{2} \right) \). Therefore, we substitute this point into the equation:
\[ \alpha^2 + \frac{1}{4} - 3\alpha + 1 = 0 \]
Simplifying the equation:
\[ \alpha^2 - 3\alpha + \frac{5}{4} = 0 \quad \Rightarrow \quad 4\alpha^2 - 12\alpha + 5 = 0 \]
Further simplifying gives us the quadratic equation:
\[ 4\alpha^2 - 10\alpha - 2\alpha + 5 = 0 \]
Factorizing the equation:
\[ (2\alpha - 1)(2\alpha - 5) = 0 \]
Solving for \( \alpha \):
\[ \alpha = \frac{1}{2} \quad \text{or} \quad \alpha = \frac{5}{2} \]
Therefore, the values of \( \alpha \) are \( \frac{1}{2} \) and \( \frac{5}{2} \).
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Concepts Used:
Circle
A circle can be geometrically defined as a combination of all the points which lie at an equal distance from a fixed point called the centre. The concepts of the circle are very important in building a strong foundation in units likes mensuration and coordinate geometry. We use circle formulas in order to calculate the area, diameter, and circumference of a circle. The length between any point on the circle and its centre is its radius.
Any line that passes through the centre of the circle and connects two points of the circle is the diameter of the circle. The radius is half the length of the diameter of the circle. The area of the circle describes the amount of space that is covered by the circle and the circumference is the length of the boundary of the circle.
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