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Q.
The locus of the foot of perpendicular drawn from focus upon a variable tangent to the parabola $2 x - y + 1^{2}=\frac{8}{\sqrt{5}}x + 2 y + 3$ is
NTA AbhyasNTA Abhyas 2022
Solution:
As we know the property of parabola i.e. the foot of the perpendicular from the focus to any tangent of a parabola lies on the tangent to the vertex.
Given the equation of parabola $\left(2 x - y + 1\right)^{2}=\frac{8 \left(x + 2 y + 3\right)}{\sqrt{5}}$
From standard equation of parabola $y^{2}=4ax$
we can say that $\left|y\right|$ is distance of a point from axis of parabola and $\left|x\right|$ is distance from tangent at vertex.
Similarly, in the give equation we can say that $2x-y+1=0$ is axis of parabola and $x+2y+3=0$ is tangent at the vertex.