Straight LineHardBloom L3
Question
Let $ax + by + c = 0$ be a variable straight line, where $a,b$ and $c$ are $1^{\text{st}\text{~}},3^{\text{rd}\text{~}}$ and $7^{\text{th}\text{~}}$ terms of some increasing A.P. Then the variable straight line always passes through a fixed point which lies on:
Options
A.$x^{2} + y^{2} = 13$
B.$x^{2} + y^{2} = 5$
C.$y^{2} = 9x$
D.$3x + 4y = 9$
Solution
Sol. [Key Concept: $a,b$ being the 1st and 3rd terms of an AP fixes a point of $ax + by + c = 0$ that lies on the given curve.]
$${ax + (a + 2d)y + (a + 6d) = 0 }{\Rightarrow a(x + y + 1) + 2d(y + 3) = 0 }{\Rightarrow (x + y + 1) + \frac{2d}{a}(y + 3) = 0 }$$∴ Fixed point $\equiv (2, - 3)$ which lies on $x^{2} + y^{2} = 13$
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