**Properties of Tangent – Parabola I**

**(iii) Length of Tangent Between the Point of Contact P(t) and the point where it meets the Directrix Q subtends right angle at focus:**

Equation of the tangent to the parabola y² = 4ax at point P(t) is ty = x + at².

It meets x = – a at (-a, (at² – a)/t)

Now, slope of SP =\({{m}_{SP}}=\frac{2at-0}{a{{t}^{2}}-a}=\frac{2t}{{{t}^{2}}-1}\).

Slope of \(SQ={{m}_{SQ}}=\frac{(a{{t}^{2}}-a)/t-0}{-a-a}=\frac{1-{{t}^{2}}}{2t}\).

Hence SP is perpendicular to SQ.

**(iv) Tangent at Extremities of Focal Chord are Perpendicular and Intersect on Directrix:**

For parabola y² = 4ax, the slope of the tangent at point P(t) is 1/t.

If PQ is a focal chord, then point Q has parameter – 1/t

Then the slope of the tangent at point Q is – t

Hence the tangents are perpendicular.

Moreover, the point of intersection of the tangent at point P(t₁) and Q(t₂) is

\((a{{t}_{1}}{{t}_{2}},a({{t}_{1}}+{{t}_{2}}))=(a\frac{1}{t}(-t),\ a(t-\frac{1}{t}))\)..

\(=(-a,a(t-\frac{1}{t}))\).

Thus, the tangents intersect on the directrix.

**(v) image of Focus in any Tangent Lies on Directrix:**

The equation of the tangent to the parabola y² = 4ax at point P(t) is ty = x + at².

It meets the y – axis at Q (0, at) Now

\({{m}_{SQ}}=\frac{at-0}{0-a}=-t\).

The equation of SQ is

y – 0 = – t (x – a)

tx + y – at = 0

Now, this line intersects the directrix x = -a at R (-a, 2at) clearly, Q (0, at) is the midpoint of SR. thus, the image of focus in any tangent lies on the directrix.