Question

From an external point P tangents are drawn to the parabola y² = 4ax, then the equation to the locus of P when these tangents makes angles q₁ and q₂ with the axis, such that

tan q₁ + tan q₂ with the axis, such that tan q₁ + tan q₂ is constant (= b), is –

• A. y = $\frac{x}{b}$
• B. y = bx
• C. y = b² x
• D. None of these

Answer 1

Answer: (B)

y = bx

Answer 2

Let the coordinates of P be (h, k) and the equation to the parabola be y² = 4ax. Any tangent on the parabola is given by y = mx + $\frac{a}{m}$. If this passes through (h, k), the coordinates will satisfy. Hence

k = mh + $\frac{a}{m}$.

Ž m²h – mk + a = 0 …(1)

Which is a quadratic in m . Let its roots be m₁ and m₂, then m₁ + m₂ = $\frac{k}{h}$ and m₁m₂ = $\frac{a}{h}$. Now, if the two tangents through P make angles q₁ and q₂ with axis of x and m₁ = tan q₁ and m₂ = tan q₂.

\ tan q₁ + tan q₂ = $\frac{k}{h}$. … (2)

and m₁m₂ = $\frac{a}{h}$ … (3)

Ž tan q₁ tan q₂ = $\frac{a}{h}$ … (4)

By hypothesis tan q₁ + tan q₂ = b

So from equation (2), $\frac{k}{h}$ = b Ž k = bh. Generalising, the locus of (h, k) is y = bx.

Hence (2) is correct answer.