Intersection of a Line and Finding Roots of a Parabola
Let S be the ...
Question
Let S be the focus of the parabola y2=8x and PQ be the common chord of the circle x2+y2−2x−4y=0 and the given parabola. The area of the ΔOPS (O is the origin) is ___.
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Solution
Parametric coordinates to y2=4ax are (at2,2at).
P(2t2,4t) should lie on x2+y2−2x−4y=0⇒4t2+16t2−4t2−16t=0⇒4t2+12t2−16t=0⇒4t(t3+3t−4)=0⇒4t(t−1)(t2+t+4)=0∴t=0,1⇒P(2,4)
Thus, area of ΔOPS=12⋅OS×PQ=12⋅(2)⋅(4)=4