A variable line y=m x+(1/2)(1-m) ∀ m ∈ R either intersects or touches curves y=(3/2)+(x+(1/2))2 and (2 y-3)2=4[(2 x+1)2-4] in total 3 points then Find absolute sum of all possible real values of m.

Question

A variable line y=m x+(1/2)(1-m) ∀ m ∈ R either intersects or touches curves y=(3/2)+(x+(1/2))2 and (2 y-3)2=4[(2 x+1)2-4] in total 3 points then Find absolute sum of all possible real values of m. (A)  (B)  (C)  (D)

Tardigrade Admin · Accepted Answer

Analyzing the given figure m ∈ 2,-2,2 √2,-2 √2 <img class="img-fluid question-image" alt="image" src="https://cdn.tardigrade.in/img/question/mathematics/0661feac6d1e5f24b039b78ecad8efc5-.jpeg" />

Q. A variable line $y = m x + \frac{1}{2} (1 - m) \forall m \in R$ either intersects or touches curves $y = \frac{3}{2} + (x + \frac{1}{2})^{2}$ and $(2 y - 3)^{2} = 4 [(2 x + 1)^{2} - 4]$ in total 3 points then Find absolute sum of all possible real values of $m$ .

Analyzing the given figure
$m \in {2, - 2, 22, - 22}$

Q. A variable line y=mx+21​(1−m)∀m∈R either intersects or touches curves y=23​+(x+21​)2 and (2y−3)2=4[(2x+1)2−4] in total 3 points then Find absolute sum of all possible real values of m.

Analyzing the given figure m∈{2,−2,22​,−22​}

Q. A variable line $y = m x + \frac{1}{2} (1 - m) \forall m \in R$ either intersects or touches curves $y = \frac{3}{2} + (x + \frac{1}{2})^{2}$ and $(2 y - 3)^{2} = 4 [(2 x + 1)^{2} - 4]$ in total 3 points then Find absolute sum of all possible real values of $m$ .

Analyzing the given figure
$m \in {2, - 2, 22, - 22}$