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If the equation of the parabola, whose vertex is at $\mathrm{\left ( 5,4 \right )}$ and the directrix is $\mathrm{3 x+y-29=0, \text { is } x^{2}+a y^{2}+b x y+c x+d y+k=0}$ , then $\mathrm{a+b+c+d+k}$ is equal to
Option: 1
$575$

Option: 2
$-575$

Option: 3
$576$

Option: 4
$-576$

Answers (1)

best_answer

Calculate foot of directrix

$\mathrm{\frac{x-5}{3}=\frac{y-4}{1}=-1 \frac{(3 \times 5+4-29)}{3^{2}+1^{2}}} \\$

$\mathrm{\Rightarrow x=8, y=5 . }$

Vertex is mid-point of focus and foot at directrix so coordinates of focus $\mathrm{=\left ( 2,3 \right )}$

Using $\mathrm{PS=PM}$ , equation of parabola is

$\mathrm{\sqrt{(x-2)^{2}+(y-3)^{2}}=\left|\frac{3 x+y-29}{\sqrt{10}}\right|}\\$

$\mathrm{\Rightarrow 10\left[x^{2}-4 x+4+y^{2}-6 y+9\right]=9 x^{2}+y^{2}+841+6 x y-58 y-174 x}\\$

$\mathrm{\Rightarrow x^{2}+9 y^{2}-6 x y+134 x-2 y-711=0 \text {. }}\\$

Compare with $\mathrm{x^{2}+a y^{2}+b x y+c x+d y+x=0}\\$

$\mathrm{a+b+c+d+k: 9-6+134-2-711=143-719}\\$

$\mathrm{= { - 576 } }$

Hence the correct answer is option 4.

Posted by

Divya Prakash Singh

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