Q.10. Property of Parabola

Question 10:
If the tangent at P on y^2=4ax meets the tangent at the vertex in Q and S is the focus of the parabola ,then the angle SQP=________?

Answer 10:
Let P = (at^2, 2at) be a parametric point on the parabola y^2 = 4ax
y^2 = 4ax => 2y dy/dx = 4a
=> dy/dx = 2a/y
=> dy/dx at P = 2a/(2at) = 1/t (slope of the tangent at P.

Eqn. of tangent at P is
y - 2at = (1/t) (x - at^2)
Tangent at the vertex is the y-axis. Hence, x-coordinate of Q = 0
Putting x= 0 in the above eqn., y = at
=> Q = (0, at) and S = (a, 0)
=> slope of SQ = (at - 0)/(0 - a) = - t
Slope of PQ x slope of SQ = (1/t) x (-t) = - 1
=> PQ is perpendicular to SQ
=> angle SQP = 90°.

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