If a line along a chord of the circle 4x2 + 4y2 + 120x + 675 = 0, passes through the point…

Question

If a line along a chord of the circle 4x2 + 4y2 + 120x + 675 = 0, passes through the point (-30, 0) and is tangent to the parabola y2 = 30x, then the length of this chord is :

Accepted Answer

Equation of tangent to y2 = 30 x

y = mx  + $${{30} \over {4m}}$$

Pass thru (-30, 0) : a = -30m + $${{30} \over {4m}}$$ $$\Rightarrow$$ m2 = 1/4

$$\Rightarrow$$ m = $${1 \over 2}$$ or m = - ${1 \over 2}$At m =${1 \over 2}$: y =${x \over 2}$+ 15$\Rightarrow$x$-$2y + 30 = 0

lAB = 2$\sqrt {{R^2} - {P^2}} $= 2$\sqrt {{{225} \over 4} - {{225} \over 5}} $$\Rightarrow$lAB = 30 .$\sqrt {{1 \over {20}}} $=${{{15} \over {\sqrt 5 }}}$= 3${\sqrt 5 }$

Solutions · 1