Straight LineHard

Question

Let PQ and MN be two straight lines touching the circle $x^{2} + y^{2} - 4x - 6y - 3 = 0$ at the points A and B respectively. Let O be the centre of the circle and $\angle AOB = \pi/3$. Then the locus of the point of intersection of the lines PQ and MN is:

Options

A.$3\left( x^{2} + y^{2} \right) - 18x - 12y + 25 = 0$
B.$x^{2} + y^{2} - 12x - 18y - 25 = 0$
C.$x^{2} + y^{2} - 18x - 12y - 25 = 0$
D.$3\left( x^{2} + y^{2} \right) - 12x - 18y - 25 = 0$

Solution

Given circle

$${x^{2} + y^{2} - 4x - 6y - 3 = 0 }{C(2,3)\& r = 4 }{cos30^{\circ} = \frac{r}{OR} = \frac{4}{OR} }{\Rightarrow OR = \frac{8}{\sqrt{3}} }$$Now

$${{OR}^{2} = (h - 2)^{2} + (k - 3)^{2} }{\Rightarrow 3\left( x^{2} + y^{2} \right) - 12x - 18y - 25 = 0}$$

Create a free account to view solution

View Solution Free
Topic: Straight Line·Practice all Straight Line questions

More Straight Line Questions

Let the direction cosines of two lines satisfy the equations: $4\mathcal{l} + m - n = 0$ and $2mn + 10n\mathcal{l} + 3\m...ABCD is a square A ≡ (1, 2), B ≡ (3, − 4). If line CD passes through (3, 8), then mid-point of CD is...Let P = (-1,0),Q = (0, 0) and R = (3, 3√3) br three points. Then, the equations of the bisector of the angle PQR i...A variable line drawn through the point (1, 3) meets the x- axis at A and y- axis at B. It the rectangle OAPB is complet...The angle between the lines represented by ax2 + 2hxy + by2 = 0 is -...