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Let C be the circle x2 + y2 + 4x - 6y - 3 = 0 and L be the locus of the point of intersection of a 2 pair of tangents to C with the angle between the two tangents equal to 60°. Then, the point at which I touches the line x = 6 is
  • a)
    (6,4)
  • b)
    (6,8)
  • c)
    (6,3)
  • d)
    (6,6)
Correct answer is option 'C'. Can you explain this answer?
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Let C be the circle x2 + y2 + 4x - 6y - 3 = 0 and L be the locus of t...
Explanation:

Given:
- Circle C: x^2 + y^2 + 4x - 6y - 3 = 0
- Locus L: Intersection of a pair of tangents to C with an angle of 60 degrees
- Point of tangency I touches the line x = 6

Finding the Point of Tangency:
1. Substitute x = 6 into the equation of circle C to find the y-coordinate of the point of tangency.
2. x = 6 gives us: 6^2 + y^2 + 4(6) - 6y - 3 = 0
3. Simplifying the equation gives: y^2 - 6y + 6 = 0
4. Solving the quadratic equation gives: y = 3 or y = 3
5. So, the point of tangency is (6, 3).
Therefore, the correct answer is option (c) (6, 3).
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Community Answer
Let C be the circle x2 + y2 + 4x - 6y - 3 = 0 and L be the locus of t...

This is the equation of a circle with radius 4 units and centered at (-2, 3)
From a point L we drop two tangents on the circle such that the angle between the tangents is 60o.

∠LPO = 90∘ 
∠PLO=30∘
 
Therefore the locus of the point L, is a circle centered at (-2, 3) and has a radius of (4 + x = 8) units.
The equation of this locus is thus, (x + 2)2 +(y − 3)2 = 82
 When x = 6, we have, (8)2 +(y − 3)2 = 82   , that is y = 3
The circle , (x + 2)2 +(y − 3)2 = 82, touches the line x = 6 at (6, 3).
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The figure below shows the street map for a certain region with the street intersections marked from a through l. A person standing at an intersection can see along straight lines to other intersections that are in her line of sight and all other people standing at these intersections. For example, a person standing at intersection g can see all people standing at intersections b, c, e, f, h, and k. In particular, the person standing at intersection g can see the person standing at intersection e irrespective of whether there is a person standing at intersection f.Six people U, V, W, X, Y, and Z, are standing at different intersections. No two people are standing at the same intersection.The following additional facts are known. 1. X, U, and Z are standing at the three corners of a triangle formed by three street segments. 2. X can see only U and Z. 3. Y can see only U and W. 4. U sees V standing in the next intersection behind Z.5. W cannot see V or Z. 6. No one among the six is standing at intersection d.Q. Who is standing at intersection a?

InstructionsComprehension:The figure below shows the street map for a certain region with the street intersections marked from a through l. A person standing at an intersection can see along straight lines to other intersections that are in her line of sight and all other people standing at these intersections. For example, a person standing at intersection g can see all people standing at intersections b, c, e, f, h, and k. In particular, the person standing at intersection g can see the person standing at intersection e irrespective of whether there is a person standing at intersection f.Six people U, V, W, X, Y, and Z, are standing at different intersections. No two people are standing at the same intersection.The following additional facts are known.1. X, U, and Z are standing at the three corners of a triangle formed by three street segments.2. X can see only U and Z.3. Y can see only U and W.4. U sees V standing in the next intersection behind Z.5. W cannot see V or Z.6. No one among the six is standing at intersection d.Q.Who is standing at intersection a?

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Let C be the circle x2 + y2 + 4x - 6y - 3 = 0 and L be the locus of the point of intersection of a 2 pair of tangents to C with the angle between the two tangents equal to 60°. Then, the point at which I touches the line x = 6 isa)(6,4)b)(6,8)c)(6,3)d)(6,6)Correct answer is option 'C'. Can you explain this answer?
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