Match the following. List-I List-II (A) The product of length of perpendicular from any point of the hyperbola x 2 − y 2 = 10 to its asymptote is. (P) 0 (B) The number of points on the ellipse 4 x 2

Question

Match the following.

List-I	List-II
(A) The product of length of perpendicular from any point of the hyperbola $x^{2} - y^{2} = 10$ to its asymptote is.	(P) $0$
(B) The number of points on the ellipse $\frac{x ^{2}}{4} + \frac{y ^{2}}{3} = 1$ from which mutually perpendicular tangents can be drawn to the hyperbola $\frac{x ^{2}}{4} - \frac{y ^{2}}{1} = 1$ is/are.	(Q) $5$
(C) The distance between the directrices of the ellipse $(4 x - 8)^{2} + 16 y^{2} = (x + 3 y + 10)^{2}$ is.	(R) $16$
(D) Tangents are drawn from any point on the line $y - x + 2 = 0$ to the parabola $y^{2} = 4 x$ such that the chords of contact pass through fixed point the sum of whose abscissa and ordinate is.	(S) $2$

TestHub · Accepted Answer

(A) Asymptotes are y = ± x ; for point ( 10 ​ sec θ , 10 ​ tan θ ) : p 1 ​ p 2 ​ = 2 10 ( s e c θ − t a n θ ) ( s e c θ + t a n θ ) ​ = 5 → Q . (B) Director circle of the hyperbola is x 2 + y 2 = 3 ; solving with the ellipse: 3 x 2 + 4 ( 3 − x 2 ) = 12 ⇒ x = 0 , y = ± 3 ​ : 2 points → S . (C) ( x − 2 ) 2 + y 2 = 4 1 ​ ( 2 x + 3 ​ y + 10 ​ ) 2 : focus (2, 0), directrix x + 3 ​ y + 10 = 0 , e = 2 1 ​ . Distance from focus to directrix = 1 + 3 ​ ∣2 + 0 + 10∣ ​ = 6 = e a ​ − a e = 2 3 a ​ ⇒ a = 4 ; distance between directrices = e 2 a ​ = 16 → R . (D) Chord of contact from ( λ , λ − 2 ) : y ( λ − 2 ) = 2 ( x + λ ) ⇒ λ ( y − 2 ) − 2 ( x + y ) = 0 , which passes through the fixed point y = 2 , x + y = 0 , i.e. ( − 2 , 2 ) : sum = 0 → P .

Mathematics - Coordinate Geometry Mixed Question with Solution | TestHub

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