A) \[{{x}^{2}}+{{y}^{2}}={{a}^{2}}\]
B) \[a{{y}^{2}}=x\]
C) \[x+a=0\]
D) \[x+y\pm a=0\]
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 \[{{m}_{1}}\,\,and\,\,{{m}_{2}},\] 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. |
You need to login to perform this action.
You will be redirected in
3 sec