CAT 2023 Slot 1 QA Question 18

First, the question says: We have a circle C defined by the equation x² + y² + 4x - 6y - 3 = 0. Then, L is the locus of the point where two tangents to C intersect, and the angle between these two tangents is 60 degrees. We need to find where L touches the line x = 6. The options are given, and the correct answer is D, which is (6, 3). But We analyze how to get there.

Okay, let's start by recalling some concepts. the equation of a circle can be written in the standard form (x - h)² + (y - k)² = r², where (h, k) is the center and r is the radius. So, I need to rewrite the given equation into this form.

The given equation is x² + y² + 4x - 6y - 3 = 0. Let me complete the squares for x and y.

For x: x² + 4x. To complete the square, I take half of 4, which is 2, square it to get 4. So, x² + 4x = (x + 2)² - 4.

For y: y² - 6y. Half of -6 is -3, squared is 9. So, y² - 6y = (y - 3)² - 9.

Now, substituting back into the equation:

(x + 2)² - 4 + (y - 3)² - 9 - 3 = 0

Simplify the constants: -4 -9 -3 = -16

So, (x + 2)² + (y - 3)² - 16 = 0

Which means (x + 2)² + (y - 3)² = 16

So, the circle C has center at (-2, 3) and radius 4.

Alright, next part. We need to find the locus L of the intersection points of two tangents to C such that the angle between the tangents is 60 degrees.

for a circle, the angle between two tangents from an external point is related to the distance from that point to the center of the circle. The formula, if I recall correctly, is:

If θ is the angle between the two tangents from an external point P, then θ = 2 arcsin(r / d), where r is the radius and d is the distance from P to the center.

So, if we have a circle with center O and radius r, and an external point P, then the angle between the two tangents from P is 2θ, where θ is the angle between OP and one of the tangents. So, using trigonometry, sinθ = r / d, so θ = arcsin(r / d). Therefore, the angle between the two tangents is 2θ = 2 arcsin(r / d).

In this problem, the angle is given as 60 degrees, which is π/3 radians. So, 60° = 2 arcsin(r / d). Let me solve for d.

Divide both sides by 2: 30° = arcsin(r / d)

So, sin(30°) = r / d

We know sin(30°) is 1/2, and the radius r is 4. So,

1/2 = 4 / d

Solving for d: d = 4 * 2 = 8

So, the distance from the external point P to the center of the circle O must be 8 units.

Therefore, the locus L is the set of all points P such that the distance from P to O is 8. That is, L is a circle centered at O with radius 8.

The center O is (-2, 3), so L is the circle (x + 2)² + (y - 3)² = 64.

But wait, is that correct? Because the locus of points from which the angle between the tangents is constant is indeed another circle, called the director circle. For a circle of radius r, the director circle is the locus of points from which the angle between the tangents is 90°, and its equation is (x - h)² + (y - k)² = 2r². But in this case, the angle is 60°, not 90°, so the radius of the director circle would be different.

Hmm, maybe I need to revisit the formula. Let me think again.

The angle between the tangents is 2θ, where θ is the angle between the line from the external point to the center and one of the tangents. So, as I had before, sinθ = r / d, so θ = arcsin(r / d). Therefore, the angle between the tangents is 2θ = 2 arcsin(r / d).

Given that angle is 60°, so 2 arcsin(r / d) = 60°, which gives arcsin(r / d) = 30°, so r / d = sin(30°) = 1/2, so d = 2r.

Wait, in our case, r is 4, so d = 8. So, the distance from P to O is 8. So, the locus L is the set of all points P such that PO = 8. So, yes, L is a circle with center O and radius 8.

So, equation of L is (x + 2)² + (y - 3)² = 64.

Now, we need to find where this circle L touches the line x = 6.

To find the point of contact, we can solve the system of equations:

Substitute x = 6 into the first equation:

(6 + 2)² + (y - 3)² = 64 8² + (y - 3)² = 64 64 + (y - 3)² = 64 Subtract 64 from both sides: (y - 3)² = 0 So, y - 3 = 0 => y = 3

Therefore, the point of contact is (6, 3), which is option D.

Wait, but let me make sure I didn't make a mistake. So, L is a circle with center (-2, 3) and radius 8. The line x = 6 is a vertical line.