A circle with centre \((2,3) \) and radius \(4\) intersects the line \(𝑥+𝑦=3\) at the points \(𝑃\) and \(𝑄\). If the tangents at \(𝑃\) and \(𝑄\) intersect at the point \(S(α,β)\), then \(4α−7β\) is equal to _____ .
Correct Answer: 11
Solution and Explanation
The given line is polar or P(2, β) w.r.t. given circle
x² + y² - 4x - 6y - 3 = 0
Chord or contact
αx + βy - 2(x + α) - 3(y + β) - 3 = 0
⇒ (α - 2)x + (β - 3)y - (2α + 3β + 3) = 0 .... (i)
⋅⋅ But the equation of chord of contact is given as: x + y - 3 = 0 .... (ii)
Comparing the coefficients:
α - 2 / 1 = β - 3 / 1 = - (2α + 3β + 3) / -3
On solving: α = -6
β = -5
Now 4α - 7β = 11
Top Questions on circle
- Let circle $C$ be the image of
$$ x^2 + y^2 - 2x + 4y - 4 = 0 $$
in the line
$$ 2x - 3y + 5 = 0 $$
and $A$ be the point on $C$ such that $OA$ is parallel to the x-axis and $A$ lies on the right-hand side of the centre $O$ of $C$.
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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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