Question

Matrix Match Type :

39. Let S = 0 be the parabola touching x-axis at (1,0) and y = x at (1,1). Let (a,b) & (p,q) be respectively focus & vertex of parabola.

Column-I

(A) Roots of the equation $x^2 - 5ax + 10b = 0$ are

(B) $\frac{p-q}{a^2}$ is equal to

(C) Roots of the equation $x^2 - 25px + 75q = 0$ are

(D) $q-4b^2$ is equal to Column-II

(P) 1

(Q) 5

(R) 12

(S) 2

Answer 1

(A) $\to$ (P), (S); (B) $\to$ (P); (C) $\to$ (P), (R); (D) $\to$ No Match

Answer 2

The equation of the parabola touching $L_1: y=0$ at $(1,0)$ and $L_2: x-y=0$ at $(1,1)$ is given by $L_1 L_2 = \lambda C^2$, where $C$ is the chord of contact $x=1$. So, $y(x-y) = \lambda(x-1)^2$. $xy - y^2 = \lambda(x^2 - 2x + 1)$. For a parabola, $h^2 = ab$. Here $a=\lambda$, $b=1$, $h=-1/2$. $(-1/2)^2 = \lambda \cdot 1 \implies \lambda = 1/4$. The equation is $\frac{1}{4}x^2 - xy + y^2 + \frac{1}{2}x - \frac{1}{4} = 0$, which simplifies to $x^2 - 4xy + 4y^2 + 2x - 1 = 0$. This can be written as $(x-2y)^2 = 2x-1$. Let $u = x-2y$ and $v = 2x+y$. Then $x = \frac{2v+u}{5}$ and $y = \frac{v-2u}{5}$. Substituting into the equation: $u^2 = 2\left(\frac{2v+u}{5}\right) - 1 = \frac{4v+2u-5}{5}$. $5u^2 = 4v+2u-5 \implies 4v = 5u^2 - 2u + 5 \implies v = \frac{5}{4}u^2 - \frac{1}{2}u + \frac{5}{4}$. The vertex in $(u,v)$ is at $u_v = -\frac{-1/2}{2(5/4)} = 1/5$ and $v_v = \frac{5}{4}(\frac{1}{5})^2 - \frac{1}{2}(\frac{1}{5}) + \frac{5}{4} = \frac{6}{5}$. Vertex $(p,q)$ in $(x,y)$ is $(\frac{2(6/5)+1/5}{5}, \frac{6/5-2(1/5)}{5}) = (13/25, 4/25)$. The focus in $(u,v)$ is $(u_v, v_v + \frac{1}{4A}) = (1/5, 6/5 + \frac{1}{4(5/4)}) = (1/5, 6/5+1/5) = (1/5, 7/5)$. Focus $(a,b)$ in $(x,y)$ is $(\frac{2(7/5)+1/5}{5}, \frac{7/5-2(1/5)}{5}) = (3/5, 1/5)$.

(A) Roots of $x^2 - 5ax + 10b = 0$: $x^2 - 5(3/5)x + 10(1/5) = 0 \implies x^2 - 3x + 2 = 0$. Roots are $1$ and $2$. Matches (P) and (S).

(B) $\frac{p-q}{a^2}$: $p=13/25, q=4/25, a=3/5$. $\frac{13/25 - 4/25}{(3/5)^2} = \frac{9/25}{9/25} = 1$. Matches (P).

(C) Roots of $x^2 - 25px + 75q = 0$: $x^2 - 25(13/25)x + 75(4/25) = 0 \implies x^2 - 13x + 12 = 0$. Roots are $1$ and $12$. Matches (P) and (R).

(D) $q-4b^2$: $q=4/25, b=1/5$. $4/25 - 4(1/5)^2 = 4/25 - 4/25 = 0$. No match in Column-II.

The question is a matrix match. Assuming the intention is to list all possible matches for roots: (A) $\to$ (P), (S) (B) $\to$ (P) (C) $\to$ (P), (R) (D) $\to$ No Match