Conic-Sections Question 551
Question: The locus of the point of intersection of two tangents to the parabola $ y^{2}=4ax, $ which are at right angle to one another is
Options:
A) $ x^{2}+y^{2}=a^{2} $
B) $ ay^{2}=x $
C) $ x+a=0 $
D) $ x+y\pm a=0 $
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Answer:
Correct Answer: C
Solution:
[c] Let the two tangents to the parabola $ y^{2}=4ax $ be PT and QT which are at right angle to one another at $ T(h,k) $ . Then we have to find the locus of T (h, k). We know that $ y=mx+\frac{a}{m}, $ where m is the slope is the equation of tangent to the parabola $ y^{2}=4ax $ for all m. Since this tangent to the parabola will pass through $ T(h,k) $ so $ k=mh+\frac{a}{m}; $ Or $ m^{2}h-mk+a=0 $ This is a quadratic equation in m so will have Two roots, say $ m_1andm_2, $ then $ m_1+m_2=\frac{k}{h}, $ and $ m_1:m_2=\frac{a}{h} $ Given that the two tangents intersect at right angle so $ m_1,m_2=-1 $ or $ \frac{a}{h}=-1 $ or $ h+a=0 $ The locus of $ T(h,k) $ is $ x+a=0, $ which is the equation of directrix.
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