If e1 and e2 are the roots of the equation x2−ax+2=0, then match the following.COLUMN - ICOLUMN - II(A) If e1 and e2 are the eccentricities of the ellipse and hyperbola, respectively then the values of a are(p) 6(B) If both e1 & e2 are the eccentricities of the hyperbolas, then values of a are(q) 52(C) If e1 and e2 are eccentricities of hyperbola and conjugate hyperbola, then values of a are(r) 22(D) If e1 is the eccentricity of the hyperbola for which there exists infinite points from which perpendicular tangents can be drawn and e2 is the eccentricity of the hyperbola in which no such points exist then the values of a are(s) 5

e1,e2 are roots of x2−ax+2=0 A) e1 0⇒f(1) 3⇒a=5,6 B) e1>1,e2>1⇒ roots greater than 1⇒Δ>0,a⋅f(1)>0⇒a∈(−α,−2)∪(2,α),a 2,e2 22

If e1 and e2 are the roots of the equation x2−ax+2=0, then match the following.COLUMN - ICOLUMN - II(A) If e1 and e2 are the eccentricities of the ellipse and hyperbola, respectively then the values of a are(p) 6(B) If both e1 & e2 are the eccentricities of the hyperbolas, then values of a are(q) 52(C) If e1 and e2 are eccentricities of hyperbola and conjugate hyperbola, then values of a are(r) 22(D) If e1 is the eccentricity of the hyperbola for which there exists infinite points from which perpendicular tangents can be drawn and e2 is the eccentricity of the hyperbola in which no such points exist then the values of a are(s) 5

e1,e2 are roots of x2−ax+2=0 A) e1 0⇒f(1) 3⇒a=5,6 B) e1>1,e2>1⇒ roots greater than 1⇒Δ>0,a⋅f(1)>0⇒a∈(−α,−2)∪(2,α),a 2,e2 22

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If e₁ and e₂ are the roots of the equation $x^{2} - ax + 2 = 0$ , then match the following.
COLUMN - I COLUMN - II
(A) If e₁ and e₂ are the eccentricities of the ellipse and hyperbola, respectively then the values of a are (p) 6
(B) If both e₁ & e₂ are the eccentricities of the hyperbolas, then values of a are (q) $\frac{5}{2}$
(C) If e₁ and e₂ are eccentricities of hyperbola and conjugate hyperbola, then values of a are (r) $2 \sqrt{2}$
(D) If e₁ is the eccentricity of the hyperbola for which there exists infinite points from which perpendicular tangents can be drawn and e₂ is the eccentricity of the hyperbola in which no such points exist then the values of a are (s) 5

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A-qs; B-q,; C-r; D-pq

A-ps; B-qr,; C-r; D-ps

A-s; B-qr,; C-s; D-ps

A-ps; B-q,; C-r; D-rs

answer is D.

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Detailed Solution

$e_{1}, e_{2} are roots of x^{2} - ax + 2 = 0$

$A) e_{1} < 1 < e_{2}, coefficient of x^{2} = 1 > 0 \Rightarrow f (1) < 0$

$\Rightarrow 1 - a + 2 < 0 \Rightarrow a > 3 \Rightarrow a = 5, 6$

$B) e_{1} > 1, e_{2} > 1 \Rightarrow roots greater than 1$

$\begin{array}{l} \Rightarrow Δ > 0, a \cdot f (1) > 0 \Rightarrow a \in (- α, - 2) \cup (2, α), a < 3 \\ \Rightarrow a \in (2, 3) \end{array}$

$C) e_{1}^{2} + e_{2}^{2} = e_{1}^{2} \cdot e_{2}^{2} \Rightarrow {(e_{1} + e_{2})}^{2} - 2 e_{1} e_{2} = {(e_{1} e_{2})}^{2}$

$\Rightarrow a^{2} - 8 = 0 \Rightarrow a = 2 \sqrt{2}$

$D) e_{1} > \sqrt{2}, e_{2} < \sqrt{2} \Rightarrow f (\sqrt{2}) < 0 \Rightarrow a > 2 \sqrt{2}$

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