Tamil Nadu Board of Secondary EducationHSC Science Class 12th

# If the normal at the point ‘t1‘ on the parabola y2 = 4ax meets the parabola again at the point ‘t2‘, then prove that t2 = tt-(t1+2t1) - Mathematics

Sum

If the normal at the point ‘t1‘ on the parabola y2 = 4ax meets the parabola again at the point ‘t2‘, then prove that t2 = - ("t"_1 + 2/"t"_1)

#### Solution

Equation of normal to y2 = 4at’ t’ is y + xt = 2at + at3.

So equation of normal at ‘t1’ is y + xt1 = 2at1 + at13

The normal meets the parabola y2 = 4ax at ‘t2

(ie.,) at (at22, 2at2)

⇒ 2at2 + at1t22 = 2at1 + at13

So 2a(t2 – t1) = at13 – at1t22

⇒ 2a(t2 – t1) = at1(t12 – t22)

⇒ 2(t2 – t1) = t1(t1 + t2)(t1 – t2)

⇒ 2= – t1(t1 + t2)

⇒ t1 + t2 = (-2)/"t"_1

⇒ t2 = - "t"_1 - 2/"t"_1

= - ("t"_1 + 2/"t"_1)

Concept: Tangents and Normals to Conics
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Chapter 5: Two Dimensional Analytical Geometry-II - Exercise 5.4 [Page 207]

#### APPEARS IN

Tamil Nadu Board Samacheer Kalvi Class 12th Mathematics Volume 1 and 2 Answers Guide
Chapter 5 Two Dimensional Analytical Geometry-II
Exercise 5.4 | Q 8 | Page 207
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