If the normal to a parabola y2=4 a x at P meets the curve again in Q and if P Q and the normal at Q make angles α and β, respectively with the x-axis, then mid tan α( tan α+ tan β) has the value equal to

Question

If the normal to a parabola y2=4 a x at P meets the curve again in Q and if P Q and the normal at Q make angles α and β, respectively with the x-axis, then mid tan α( tan α+ tan β) has the value equal to (A)  (B)  (C)  (D)

Tardigrade Admin · Accepted Answer

tan α=-t1 and tan β=-t2 also t2=-t1-(2/t1) <img class="img-fluid question-image" alt="image" src="https://cdn.tardigrade.in/img/question/mathematics/329b09a53714600ae858a5cecc0d3ca2-.png" /> t1 t2+t12=-2 tan α tan β+ tan 2 α=-2 ⇒ (B).

Q. If the normal to a parabola $y^{2} = 4 a x$ at $P$ meets the curve again in $Q$ and if $PQ$ and the normal at $Q$ make angles $α$ and $β$ , respectively with the $x$ -axis, then $∣ tan α (tan α +$ $tan β)$ has the value equal to

$tan α = - t_{1}$ and $tan β = - t_{2}$
also $t_{2} = - t_{1} - \frac{2}{t _{1}}$

$t_{1} t_{2} + t_{1}^{2} = - 2$
$tan α tan β + tan^{2} α = - 2 \Rightarrow (B)$ .

Q. If the normal to a parabola y2=4ax at P meets the curve again in Q and if PQ and the normal at Q make angles α and β, respectively with the x-axis, then ∣tanα(tanα+ tanβ) has the value equal to

tanα=−t1​ and tanβ=−t2​ also t2​=−t1​−t1​2​ t1​t2​+t12​=−2 tanαtanβ+tan2α=−2⇒(B).

Q. If the normal to a parabola $y^{2} = 4 a x$ at $P$ meets the curve again in $Q$ and if $PQ$ and the normal at $Q$ make angles $α$ and $β$ , respectively with the $x$ -axis, then $∣ tan α (tan α +$ $tan β)$ has the value equal to

$tan α = - t_{1}$ and $tan β = - t_{2}$
also $t_{2} = - t_{1} - \frac{2}{t _{1}}$

$t_{1} t_{2} + t_{1}^{2} = - 2$
$tan α tan β + tan^{2} α = - 2 \Rightarrow (B)$ .