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Question

If the line segment intercepted by the parabola $${ y }^{ 2 }=4\alpha x$$ on the line $$ax+by+c=0$$ subtends a right angle at the vertex, then 


A
4aα+c=0
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B
4bα+c=0
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C
4aα+b=0
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D
none of these
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Solution

The correct option is B $$4a\alpha +c=0$$
Making the equation of parabola $${ y }^{ 2 }=4\alpha x$$ homogeneous using the equation of line $$ax+by+c=0$$, we get
$$\displaystyle { y }^{ 2 }=4\alpha x\left( \frac { ax+by }{ -c }  \right) \Rightarrow 4a\alpha { x }^{ 2 }+4b\alpha xy+c{ y }^{ 2 }=0$$
which represents the combined equation of $$OP$$ and $$OQ$$.
Since $$\angle POQ={ 90 }^{ 0 }$$, coefficients of $${ x }^{ 2 }+$$ coefficients of $${ y }^{ 2 }=0$$
$$\Rightarrow 4a\alpha +c=0$$

390907_135488_ans_666d35d81ec2414686c35f812d270588.png

Mathematics

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