Let P1 be a parabola with vertex (3, 2) and focus (4, 4) and P2 be its mirror image with respect to the line x + 2y = 6. Then the directrix of P2 is x + 2y = ____________.
Answer (integer)
10
Solution
<p>Focus = (4, 4) and vertex = (3, 2)</p>
<p>$\therefore$ Point of intersection of directrix with axis of parabola = A = (2, 0)</p>
<p>Image of A(2, 0) with respect to line x + 2y = 6 is B(x<sub>2</sub>, y<sub>2</sub>)</p>
<p>$\therefore$ ${{{x_2} - 2} \over 1} = {{{y_2} - 0} \over 2} = {{ - 2(2 + 0 - 6)} \over 5}$</p>
<p>$\therefore$ $B({x_2},\,{y_2}) = \left( {{{18} \over 5},{{16} \over 5}} \right)$.</p>
<p>Point B is point of intersection of direction with axes of parabola P<sub>2</sub>.</p>
<p>$\therefore$ $x + 2y = \lambda$ must have point $\left( {{{18} \over 5},{{16} \over 5}} \right)$</p>
<p>$\therefore$ $x + 2y = 10$</p>
About this question
Subject: Mathematics · Chapter: Conic Sections · Topic: Parabola
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