If P is a point on the circle , Q is a point on the straight line and is the perpendicular bisector of PQ, then 13 times the sum of abscissa of all such points P is _________

Question

If P is a point on the circle $x^{2} + y^{2} = 4,$ , Q is a point on the straight line $5 x + y + 2 = 0$ and $x - y + 1 = 0$ is the perpendicular bisector of PQ, then 13 times the sum of abscissa of all such points P is __________. [2026]

Smart Achievers · Accepted Answer

(2)

Mid point of $P Q$ lies on $x - y + 1 = 0$

$\frac{2 \cos θ + α}{2} - \frac{2 \sin θ - 5 α - 2}{2} + 1 = 0$

$2 \cos θ + α - 2 \sin θ + 5 α + 2 + 2 = 0$

$\cos θ - \sin θ + 3 α + 2 = 0 . . . (1)$

$∵$ $Slope of P Q is - 1$

$\frac{2 \sin θ + 5 α + 2}{2 \cos θ - α} = - 1$

$2 \sin θ + 5 α + 2 = - 2 \cos θ + α$

$\sin θ + \cos θ + 2 α + 1 = 0 . . . (2)$

Eliminate $α$ from (1) and (2)

$\Rightarrow \cos θ + 5 \sin θ = 1, θ \in [0, 2 π]$

$\Rightarrow 5 \times 2 \sin \frac{θ}{2} \cos \frac{θ}{2} = 2 \sin^{2} \frac{θ}{2}$

$∴$ $\sin \frac{θ}{2} = 0 \Rightarrow \cos θ = 1$

or

$\sin \frac{θ}{2} = 5 \Rightarrow \cos θ = - \frac{12}{13}$

Sum of all possible values of abscissa of point $P$ is

$= 2 \times 1 + 2 (\frac{- 12}{13}) = \frac{2}{13}$

$∴$ $13 times sum of all possible values of abscissa of point P is = 2$

Solution

Q.
If P is a point on the circle $x^{2} + y^{2} = 4,$ , Q is a point on the straight line $5 x + y + 2 = 0$ and $x - y + 1 = 0$ is the perpendicular bisector of PQ, then 13 times the sum of abscissa of all such points P is __________. [2026]

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Solution

Q. If P is a point on the circle x2+y2=4,, Q is a point on the straight line 5x+y+2=0 and x−y+1=0 is the perpendicular bisector of PQ, then 13 times the sum of abscissa of all such points P is __________. [2026]

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Q.
If P is a point on the circle $x^{2} + y^{2} = 4,$ , Q is a point on the straight line $5 x + y + 2 = 0$ and $x - y + 1 = 0$ is the perpendicular bisector of PQ, then 13 times the sum of abscissa of all such points P is __________. [2026]