Question

The vertex of the parabola whose focus is(2,1) & directrix is x-2y+10=0 is

Answer 1

EXPLANATION.

=> vertex of parabola whose focus is ( 2,1)

=> directrix is = x - 2y + 10 = 0

=> Let p(x, y) be any point on the parabola

whose focus is s =(2,1).

=> draw PM perpendicular to the directrix

x - 2y + 10 = 0.

=> As we know that.

=> SP = PM

=> SP² = PM²

$\sf : \implies \: (x - 2) {}^{2} + (y - 1) {}^{2} = ( \dfrac{x - 2y + 10}{ \sqrt{1 + 4} } ) {}^{2} \\ \\ \sf : \implies \: (x - 2) {}^{2} + (y - 1) {}^{2} = \frac{(x - 2y + 10) {}^{2} }{5} \\ \\ \sf : \implies \: {x}^{2} + 4 - 4x + ( {y}^{2} + 1 - 2y) = \frac{(x - 2y + 10) {}^{2} }{5} \\ \\ \sf : \implies \: 5( {x}^{2} + {y}^{2} - 4x - 2y + 5) = (x - 2y + 10) {}^{2}$

$\sf : \implies \: 5 {x}^{2} + 5 {y}^{2} - 20x - 10y + 25 = (x - 2y + 10) {}^{2} \\ \\ \sf : \implies \: {5x}^{2} + 5 {y}^{2} - 20x - 10y + 25 = {x}^{2} + 4 {y}^{2} + 100 - 4xy - 40y + 20x \\ \\ \sf : \implies \: 4 {x}^{2} + {y}^{2} - 40x + 20y + 4xy - 75 = 0$