Let the points P and Q lie on the lines y=x and y=2x respectively. The equation of the locus of the mid-point of the line segment joining P and Q, if |P Q|=4, is

Question

Let the points P and Q lie on the lines y=x and y=2x respectively. The equation of the locus of the mid-point of the line segment joining P and Q, if |P Q|=4, is (A) 25x2+36xy+13y2=4 (B) 25x2-36xy+13y2=4 (C) 25x2-36xy-13y2=4 (D) 25x2+36xy-13y2=4

Tardigrade Admin · Accepted Answer

<img class="img-fluid question-image" alt="Solution" src="https://cdn.tardigrade.in/q/nta/m-nnkdjdelnqfq.png" /> Let, P(a , a) and Q(b , 2 b) and the midpoint of P Q is (h , k) 2h=a+b 2k=a+2b a=2(2 h - k) and b=2(k - h) Now, (a - b)2+(a - 2 b)2=16 ⇒ (6 h - 4 k)2+(8 h - 6 k)2=16 ⇒ 9h2+4k2-12hk+16h2+9k2-24hk=4 ⇒ 25x2+13y2-36xy=4

Q. Let the points $P$ and $Q$ lie on the lines $y = x$ and $y = 2 x$ respectively. The equation of the locus of the mid-point of the line segment joining $P$ and $Q,$ if $∣ PQ ∣ = 4,$ is

$25 x^{2} + 36 x y + 13 y^{2} = 4$

$25 x^{2} - 36 x y + 13 y^{2} = 4$

$25 x^{2} - 36 x y - 13 y^{2} = 4$

$25 x^{2} + 36 x y - 13 y^{2} = 4$

Q. Let the points P and Q lie on the lines y=x and y=2x respectively. The equation of the locus of the mid-point of the line segment joining P and Q, if ∣PQ∣=4, is

25x2+36xy+13y2=4

25x2−36xy+13y2=4

25x2−36xy−13y2=4

25x2+36xy−13y2=4

Let, P(a,a) and Q(b,2b) and the midpoint of P Q is (h,k) 2h=a+b 2k=a+2b a=2(2h−k) and b=2(k−h) Now, (a−b)2+(a−2b)2=16 ⇒(6h−4k)2+(8h−6k)2=16 ⇒9h2+4k2−12hk+16h2+9k2−24hk=4 ⇒25x2+13y2−36xy=4

Q. Let the points $P$ and $Q$ lie on the lines $y = x$ and $y = 2 x$ respectively. The equation of the locus of the mid-point of the line segment joining $P$ and $Q,$ if $∣ PQ ∣ = 4,$ is

$25 x^{2} + 36 x y + 13 y^{2} = 4$

$25 x^{2} - 36 x y + 13 y^{2} = 4$

$25 x^{2} - 36 x y - 13 y^{2} = 4$

$25 x^{2} + 36 x y - 13 y^{2} = 4$